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<!DOCTYPE html><html> <head> <title>maximally consistent</title>  <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <link rel="stylesheet" href="LaTeXML.css" type="text/css"> <link rel="stylesheet" href="ltx-article.css" type="text/css"> <link rel="stylesheet" href="https://cdn.rawgit.com/holtzermann17/3f71ceeb3b055e1ddc3b6c11fb1f074c/raw/2bb23e3b173ff96840797fc0c3bcb8c54085df8e/LaTeXML.css" type="text/css"> <link rel="stylesheet" href="https://cdn.rawgit.com/holtzermann17/4bda0365b30858ac2fb83623185fe3ec/raw/cedd84ed3e3ad597c5d293f443ecfe4803741c6b/ltx-article.css" type="text/css"> <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML" type="text/javascript"></script> </head> <body> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line"> <h1 class="ltx_title ltx_title_document">maximally consistent</h1> <div id="p1" class="ltx_para"> <br class="ltx_break"> <p class="ltx_p">A set <math id="p1.m1" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> of <a class="nnexus_concept" href="http://planetmath.org/word">well-formed formulas</a> (wff) is <a class="nnexus_concept" href="http://planetmath.org/maximallyconsistent">maximally consistent</a> if <math id="p1.m2" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is <a class="nnexus_concept" href="http://planetmath.org/consistent">consistent</a> and any consistent <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">superset</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Superset.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/superset"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> of it is itself: <math id="p1.m3" class="ltx_Math" alttext="\Delta\subseteq\Gamma" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊆</mo><mi mathvariant="normal">Γ</mi></mrow></math> with <math id="p1.m4" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math> consistent implies <math id="p1.m5" class="ltx_Math" alttext="\Gamma=\Delta" display="inline"><mrow><mi mathvariant="normal">Γ</mi><mo>=</mo><mi mathvariant="normal">Δ</mi></mrow></math>.</p> </div> <div id="p2" class="ltx_para"> <p class="ltx_p">Below are some basic <a class="nnexus_concept" href="http://planetmath.org/property">properties</a> of a maximally consistent set <math id="p2.m1" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math>:</p> <ol id="I1" class="ltx_enumerate"> <li id="I1.i1" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">1.</span> <div id="I1.i1.p1" class="ltx_para"> <p class="ltx_p"><math id="I1.i1.p1.m1" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is <a class="nnexus_concept" href="http://planetmath.org/firstordertheory">deductively closed</a> (<math id="I1.i1.p1.m2" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is a theory): <math id="I1.i1.p1.m3" class="ltx_Math" alttext="\Delta\vdash A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mi>A</mi></mrow></math> iff <math id="I1.i1.p1.m4" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>.</p> </div> </li> <li id="I1.i2" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">2.</span> <div id="I1.i2.p1" class="ltx_para"> <p class="ltx_p"><math id="I1.i2.p1.m1" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is <em class="ltx_emph ltx_font_italic"><a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">complete</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/ordersinanumberfield"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/soundcomplete"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/completebinarytree"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/completegraph"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/kripkesemantics"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup></em>: <math id="I1.i2.p1.m2" class="ltx_Math" alttext="\Delta\vdash A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mi>A</mi></mrow></math> or <math id="I1.i2.p1.m3" class="ltx_Math" alttext="\Delta\vdash\neg A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow></mrow></math> for any wff <math id="I1.i2.p1.m4" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math>.</p> </div> </li> <li id="I1.i3" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">3.</span> <div id="I1.i3.p1" class="ltx_para"> <p class="ltx_p">for any wff <math id="I1.i3.p1.m1" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math>, either <math id="I1.i3.p1.m2" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> or <math id="I1.i3.p1.m3" class="ltx_Math" alttext="\neg A\in\Delta" display="inline"><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>.</p> </div> </li> <li id="I1.i4" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">4.</span> <div id="I1.i4.p1" class="ltx_para"> <p class="ltx_p">If <math id="I1.i4.p1.m1" class="ltx_Math" alttext="A\notin\Delta" display="inline"><mrow><mi>A</mi><mo>∉</mo><mi mathvariant="normal">Δ</mi></mrow></math>, then <math id="I1.i4.p1.m2" class="ltx_Math" alttext="\Delta\cup\{A\}" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>∪</mo><mrow><mo stretchy="false">{</mo><mi>A</mi><mo stretchy="false">}</mo></mrow></mrow></math> is not consistent.</p> </div> </li> <li id="I1.i5" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">5.</span> <div id="I1.i5.p1" class="ltx_para"> <p class="ltx_p"><math id="I1.i5.p1.m1" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is a logic: <math id="I1.i5.p1.m2" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> contains all <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">theorems</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Theorem.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/lemma"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> and is <a class="nnexus_concept" href="http://planetmath.org/operation">closed under</a> <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">modus ponens</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/ModusPonens.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/modusponens"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>.</p> </div> </li> <li id="I1.i6" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">6.</span> <div id="I1.i6.p1" class="ltx_para"> <p class="ltx_p"><math id="I1.i6.p1.m1" class="ltx_Math" alttext="\perp\notin\Delta" display="inline"><mrow><mo>⟂</mo><mo>∉</mo><mi mathvariant="normal">Δ</mi></mrow></math>.</p> </div> </li> <li id="I1.i7" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">7.</span> <div id="I1.i7.p1" class="ltx_para"> <p class="ltx_p"><math id="I1.i7.p1.m1" class="ltx_Math" alttext="A\to B\in\Delta" display="inline"><mrow><mi>A</mi><mo>→</mo><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> iff <math id="I1.i7.p1.m2" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> implies <math id="I1.i7.p1.m3" class="ltx_Math" alttext="B\in\Delta" display="inline"><mrow><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>.</p> </div> </li> <li id="I1.i8" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">8.</span> <div id="I1.i8.p1" class="ltx_para"> <p class="ltx_p"><math id="I1.i8.p1.m1" class="ltx_Math" alttext="A\land B\in\Delta" display="inline"><mrow><mrow><mi>A</mi><mo>∧</mo><mi>B</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> iff <math id="I1.i8.p1.m2" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> and <math id="I1.i8.p1.m3" class="ltx_Math" alttext="B\in\Delta" display="inline"><mrow><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>.</p> </div> </li> <li id="I1.i9" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">9.</span> <div id="I1.i9.p1" class="ltx_para"> <p class="ltx_p"><math id="I1.i9.p1.m1" class="ltx_Math" alttext="A\lor B\in\Delta" display="inline"><mrow><mrow><mi>A</mi><mo>∨</mo><mi>B</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> iff <math id="I1.i9.p1.m2" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> or <math id="I1.i9.p1.m3" class="ltx_Math" alttext="B\in\Delta" display="inline"><mrow><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>.</p> </div> </li> </ol> </div> <div class="ltx_proof"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div id="p3" class="ltx_para"> <ol id="I2" class="ltx_enumerate"> <li id="I2.i1" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">1.</span> <div id="I2.i1.p1" class="ltx_para"> <p class="ltx_p">If <math id="I2.i1.p1.m1" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>, then clearly <math id="I2.i1.p1.m2" class="ltx_Math" alttext="\Delta\vdash A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mi>A</mi></mrow></math>. <a class="nnexus_concept" href="http://planetmath.org/converse">Conversely</a>, suppose <math id="I2.i1.p1.m3" class="ltx_Math" alttext="\Delta\vdash A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mi>A</mi></mrow></math>. Let <math id="I2.i1.p1.m4" class="ltx_Math" alttext="\mathcal{E}" display="inline"><mi class="ltx_font_mathcaligraphic">ℰ</mi></math> be a <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">deduction</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Deduction.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/deduction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/derivationsinnaturaldeduction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> of <math id="I2.i1.p1.m5" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math> from <math id="I2.i1.p1.m6" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math>, and <math id="I2.i1.p1.m7" class="ltx_Math" alttext="\Gamma:=\Delta\cup\{A\}" display="inline"><mrow><mi mathvariant="normal">Γ</mi><mo>:=</mo><mrow><mi mathvariant="normal">Δ</mi><mo>∪</mo><mrow><mo stretchy="false">{</mo><mi>A</mi><mo stretchy="false">}</mo></mrow></mrow></mrow></math>. Suppose <math id="I2.i1.p1.m8" class="ltx_Math" alttext="\Gamma\vdash B" display="inline"><mrow><mi mathvariant="normal">Γ</mi><mo>⊢</mo><mi>B</mi></mrow></math>. Let <math id="I2.i1.p1.m9" class="ltx_Math" alttext="\mathcal{E}_{1}" display="inline"><msub><mi class="ltx_font_mathcaligraphic">ℰ</mi><mn>1</mn></msub></math> be a deduction of <math id="I2.i1.p1.m10" class="ltx_Math" alttext="B" display="inline"><mi>B</mi></math> from <math id="I2.i1.p1.m11" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math>, then <math id="I2.i1.p1.m12" class="ltx_Math" alttext="\mathcal{E},\mathcal{E}_{1}" display="inline"><mrow><mi class="ltx_font_mathcaligraphic">ℰ</mi><mo>,</mo><msub><mi class="ltx_font_mathcaligraphic">ℰ</mi><mn>1</mn></msub></mrow></math> is a deduction of <math id="I2.i1.p1.m13" class="ltx_Math" alttext="B" display="inline"><mi>B</mi></math> from <math id="I2.i1.p1.m14" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math>, so <math id="I2.i1.p1.m15" class="ltx_Math" alttext="\Delta\vdash B" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mi>B</mi></mrow></math>. Since <math id="I2.i1.p1.m16" class="ltx_Math" alttext="\Delta\not\vdash\perp" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊬</mo><mo>⟂</mo></mrow></math>, <math id="I2.i1.p1.m17" class="ltx_Math" alttext="\Gamma\not\vdash\perp" display="inline"><mrow><mi mathvariant="normal">Γ</mi><mo>⊬</mo><mo>⟂</mo></mrow></math>, so <math id="I2.i1.p1.m18" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math> is consistent. Since <math id="I2.i1.p1.m19" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is maximal, <math id="I2.i1.p1.m20" class="ltx_Math" alttext="\Gamma=\Delta" display="inline"><mrow><mi mathvariant="normal">Γ</mi><mo>=</mo><mi mathvariant="normal">Δ</mi></mrow></math>, or <math id="I2.i1.p1.m21" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>.</p> </div> </li> <li id="I2.i2" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">2.</span> <div id="I2.i2.p1" class="ltx_para"> <p class="ltx_p">Suppose <math id="I2.i2.p1.m1" class="ltx_Math" alttext="\Delta\not\vdash A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊬</mo><mi>A</mi></mrow></math>, then <math id="I2.i2.p1.m2" class="ltx_Math" alttext="A\notin\Delta" display="inline"><mrow><mi>A</mi><mo>∉</mo><mi mathvariant="normal">Δ</mi></mrow></math> by 1. Then <math id="I2.i2.p1.m3" class="ltx_Math" alttext="\Delta\cup\{A\}" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>∪</mo><mrow><mo stretchy="false">{</mo><mi>A</mi><mo stretchy="false">}</mo></mrow></mrow></math> is not consistent (since <math id="I2.i2.p1.m4" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is maximal), which means <math id="I2.i2.p1.m5" class="ltx_Math" alttext="\Delta,A\vdash\perp" display="inline"><mrow><mrow><mi mathvariant="normal">Δ</mi><mo>,</mo><mi>A</mi></mrow><mo>⊢</mo><mo>⟂</mo></mrow></math>, or <math id="I2.i2.p1.m6" class="ltx_Math" alttext="\Delta\vdash A\to\perp" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mrow><mi>A</mi><mo>→</mo><mo>⟂</mo></mrow></mrow></math>, or <math id="I2.i2.p1.m7" class="ltx_Math" alttext="\Delta\vdash\neg A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow></mrow></math>.</p> </div> </li> <li id="I2.i3" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">3.</span> <div id="I2.i3.p1" class="ltx_para"> <p class="ltx_p">If <math id="I2.i3.p1.m1" class="ltx_Math" alttext="A\notin\Delta" display="inline"><mrow><mi>A</mi><mo>∉</mo><mi mathvariant="normal">Δ</mi></mrow></math>, then <math id="I2.i3.p1.m2" class="ltx_Math" alttext="\Delta\not\vdash A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊬</mo><mi>A</mi></mrow></math> by 1, so <math id="I2.i3.p1.m3" class="ltx_Math" alttext="\Delta\vdash\neg A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow></mrow></math> by 2, and therefore <math id="I2.i3.p1.m4" class="ltx_Math" alttext="\neg A\in\Delta" display="inline"><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> by 1 again.</p> </div> </li> <li id="I2.i4" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">4.</span> <div id="I2.i4.p1" class="ltx_para"> <p class="ltx_p">If <math id="I2.i4.p1.m1" class="ltx_Math" alttext="A\notin\Delta" display="inline"><mrow><mi>A</mi><mo>∉</mo><mi mathvariant="normal">Δ</mi></mrow></math>, then <math id="I2.i4.p1.m2" class="ltx_Math" alttext="\neg A\in\Delta" display="inline"><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> by 3., so that <math id="I2.i4.p1.m3" class="ltx_Math" alttext="\neg A,A,\perp" display="inline"><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow><mo>,</mo><mi>A</mi><mo>,</mo><mo>⟂</mo></mrow></math> is a deduction of <math id="I2.i4.p1.m4" class="ltx_Math" alttext="\perp" display="inline"><mo>⟂</mo></math> from <math id="I2.i4.p1.m5" class="ltx_Math" alttext="\Delta\cup\{A\}" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>∪</mo><mrow><mo stretchy="false">{</mo><mi>A</mi><mo stretchy="false">}</mo></mrow></mrow></math>, showing that <math id="I2.i4.p1.m6" class="ltx_Math" alttext="\Delta\cup\{A\}" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>∪</mo><mrow><mo stretchy="false">{</mo><mi>A</mi><mo stretchy="false">}</mo></mrow></mrow></math> is not consistent.</p> </div> </li> <li id="I2.i5" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">5.</span> <div id="I2.i5.p1" class="ltx_para"> <p class="ltx_p">If <math id="I2.i5.p1.m1" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math> is a theorem, then <math id="I2.i5.p1.m2" class="ltx_Math" alttext="\Delta\vdash A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mi>A</mi></mrow></math>, so that <math id="I2.i5.p1.m3" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> by 1. If <math id="I2.i5.p1.m4" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> and <math id="I2.i5.p1.m5" class="ltx_Math" alttext="A\to B\in\Delta" display="inline"><mrow><mi>A</mi><mo>→</mo><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>, then <math id="I2.i5.p1.m6" class="ltx_Math" alttext="A,A\to B,B" display="inline"><mrow><mrow><mrow><mi>A</mi><mo>,</mo><mi>A</mi></mrow><mo>→</mo><mi>B</mi></mrow><mo>,</mo><mi>B</mi></mrow></math> is a deduction of <math id="I2.i5.p1.m7" class="ltx_Math" alttext="B" display="inline"><mi>B</mi></math> from <math id="I2.i5.p1.m8" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math>, so <math id="I2.i5.p1.m9" class="ltx_Math" alttext="B\in\Delta" display="inline"><mrow><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> by 1.</p> </div> </li> <li id="I2.i6" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">6.</span> <div id="I2.i6.p1" class="ltx_para"> <p class="ltx_p">This is true for any consistent set.</p> </div> </li> <li id="I2.i7" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">7.</span> <div id="I2.i7.p1" class="ltx_para"> <p class="ltx_p">Suppose <math id="I2.i7.p1.m1" class="ltx_Math" alttext="A\to B\in\Delta" display="inline"><mrow><mi>A</mi><mo>→</mo><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>. If <math id="I2.i7.p1.m2" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>, then <math id="I2.i7.p1.m3" class="ltx_Math" alttext="B\in\Delta" display="inline"><mrow><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> since <math id="I2.i7.p1.m4" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is closed under modus ponens. Conversely, suppose <math id="I2.i7.p1.m5" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> implies <math id="I2.i7.p1.m6" class="ltx_Math" alttext="B\in\Delta" display="inline"><mrow><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>. This means that <math id="I2.i7.p1.m7" class="ltx_Math" alttext="\Delta,A\vdash B" display="inline"><mrow><mrow><mi mathvariant="normal">Δ</mi><mo>,</mo><mi>A</mi></mrow><mo>⊢</mo><mi>B</mi></mrow></math>. Then <math id="I2.i7.p1.m8" class="ltx_Math" alttext="\Delta\vdash A\to B" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow></mrow></math> by the <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">deduction theorem</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/DeductionTheorem.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/deductiontheorem"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>, and therefore <math id="I2.i7.p1.m9" class="ltx_Math" alttext="A\to B\in\Delta" display="inline"><mrow><mi>A</mi><mo>→</mo><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> by 1. </p> </div> </li> <li id="I2.i8" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">8.</span> <div id="I2.i8.p1" class="ltx_para"> <p class="ltx_p">Suppose <math id="I2.i8.p1.m1" class="ltx_Math" alttext="A\land B\in\Delta" display="inline"><mrow><mrow><mi>A</mi><mo>∧</mo><mi>B</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>, then by modus ponens on theorems <math id="I2.i8.p1.m2" class="ltx_Math" alttext="A\land B\to A" display="inline"><mrow><mrow><mi>A</mi><mo>∧</mo><mi>B</mi></mrow><mo>→</mo><mi>A</mi></mrow></math> and <math id="I2.i8.p1.m3" class="ltx_Math" alttext="A\land B\to B" display="inline"><mrow><mrow><mi>A</mi><mo>∧</mo><mi>B</mi></mrow><mo>→</mo><mi>B</mi></mrow></math>, we get <math id="I2.i8.p1.m4" class="ltx_Math" alttext="A,B\in\Delta" display="inline"><mrow><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>, since <math id="I2.i8.p1.m5" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is a logic by 5. Conversely, suppose <math id="I2.i8.p1.m6" class="ltx_Math" alttext="A,B\in\Delta" display="inline"><mrow><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>, then by modus ponens twice on theorem <math id="I2.i8.p1.m7" class="ltx_Math" alttext="A\to(B\to A\land B)" display="inline"><mrow><mi>A</mi><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>→</mo><mi>A</mi><mo>∧</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></mrow></math>, we get <math id="I2.i8.p1.m8" class="ltx_Math" alttext="A\land B\in\Delta" display="inline"><mrow><mrow><mi>A</mi><mo>∧</mo><mi>B</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> by 5.</p> </div> </li> <li id="I2.i9" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">9.</span> <div id="I2.i9.p1" class="ltx_para"> <p class="ltx_p">Suppose <math id="I2.i9.p1.m1" class="ltx_Math" alttext="A\lor B\in\Delta" display="inline"><mrow><mrow><mi>A</mi><mo>∨</mo><mi>B</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>. Then <math id="I2.i9.p1.m2" class="ltx_Math" alttext="\neg(\neg A\land\neg B)\in\Delta" display="inline"><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow><mo>∧</mo><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>B</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> by the <a class="nnexus_concept" href="http://planetmath.org/definition">definition</a> of <math id="I2.i9.p1.m3" class="ltx_Math" alttext="\lor" display="inline"><mo>∨</mo></math>, so <math id="I2.i9.p1.m4" class="ltx_Math" alttext="\neg A\land\neg B\notin\Delta" display="inline"><mrow><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow><mo>∧</mo><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>B</mi></mrow></mrow><mo>∉</mo><mi mathvariant="normal">Δ</mi></mrow></math> by 3., which means <math id="I2.i9.p1.m5" class="ltx_Math" alttext="\neg A\notin\Delta" display="inline"><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow><mo>∉</mo><mi mathvariant="normal">Δ</mi></mrow></math> or <math id="I2.i9.p1.m6" class="ltx_Math" alttext="\neg B\notin\Delta" display="inline"><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>B</mi></mrow><mo>∉</mo><mi mathvariant="normal">Δ</mi></mrow></math> by the <a class="nnexus_concept" href="http://planetmath.org/contrapositive">contrapositive</a> of 8, or <math id="I2.i9.p1.m7" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> or <math id="I2.i9.p1.m8" class="ltx_Math" alttext="B\in\Delta" display="inline"><mrow><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> by 3. Conversely, suppose <math id="I2.i9.p1.m9" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> or <math id="I2.i9.p1.m10" class="ltx_Math" alttext="B\in\Delta" display="inline"><mrow><mi>B</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math>. Then by modus ponens on theorems <math id="I2.i9.p1.m11" class="ltx_Math" alttext="A\to A\lor B" display="inline"><mrow><mi>A</mi><mo>→</mo><mrow><mi>A</mi><mo>∨</mo><mi>B</mi></mrow></mrow></math> or <math id="I2.i9.p1.m12" class="ltx_Math" alttext="B\to A\lor B" display="inline"><mrow><mi>B</mi><mo>→</mo><mrow><mi>A</mi><mo>∨</mo><mi>B</mi></mrow></mrow></math> respectively, we get <math id="I2.i9.p1.m13" class="ltx_Math" alttext="A\lor B\in\Delta" display="inline"><mrow><mrow><mi>A</mi><mo>∨</mo><mi>B</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> by 5.</p> </div> </li> </ol> <p class="ltx_p">∎</p> </div> </div> <div id="p4" class="ltx_para"> <p class="ltx_p">The <a class="nnexus_concept" href="http://mathworld.wolfram.com/Converse.html">converses</a> of 2 and 3 above are true too, and they provide alternative definitions of maximal <a class="nnexus_concept" href="http://mathworld.wolfram.com/Consistency.html">consistency</a>.</p> <ol id="I3" class="ltx_enumerate"> <li id="I3.i1" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">1.</span> <div id="I3.i1.p1" class="ltx_para"> <p class="ltx_p">any complete consistent theory is maximally consistent.</p> </div> </li> <li id="I3.i2" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_enumerate">2.</span> <div id="I3.i2.p1" class="ltx_para"> <p class="ltx_p">any consistent set satisfying the condition in 3 above is maximally consistent.</p> </div> </li> </ol> </div> <div class="ltx_proof"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div id="p5" class="ltx_para"> <p class="ltx_p">Suppose <math id="p5.m1" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is complete consistent. Let <math id="p5.m2" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math> be a consistent superset of <math id="p5.m3" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math>. <math id="p5.m4" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math> is also complete. If <math id="p5.m5" class="ltx_Math" alttext="A\in\Gamma-\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mrow><mi mathvariant="normal">Γ</mi><mo>-</mo><mi mathvariant="normal">Δ</mi></mrow></mrow></math>, then <math id="p5.m6" class="ltx_Math" alttext="\Gamma\vdash A" display="inline"><mrow><mi mathvariant="normal">Γ</mi><mo>⊢</mo><mi>A</mi></mrow></math>, so <math id="p5.m7" class="ltx_Math" alttext="\Gamma\not\vdash\neg A" display="inline"><mrow><mi mathvariant="normal">Γ</mi><mo>⊬</mo><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow></mrow></math> since <math id="p5.m8" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math> is consistent. But then <math id="p5.m9" class="ltx_Math" alttext="\Delta\not\vdash\neg A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊬</mo><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow></mrow></math> since <math id="p5.m10" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math> is a superset of <math id="p5.m11" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math>, which means <math id="p5.m12" class="ltx_Math" alttext="\Delta\vdash A" display="inline"><mrow><mi mathvariant="normal">Δ</mi><mo>⊢</mo><mi>A</mi></mrow></math> since <math id="p5.m13" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is complete. But then <math id="p5.m14" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> since <math id="p5.m15" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is deductively closed, which is a <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">contradiction</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Contradiction.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/contradiction"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/contradictorystatement"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup>. Hence <math id="p5.m16" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is maximal.</p> </div> <div id="p6" class="ltx_para"> <p class="ltx_p">Next, suppose <math id="p6.m1" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> is consistent satisfying the condition: either <math id="p6.m2" class="ltx_Math" alttext="A\in\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> or <math id="p6.m3" class="ltx_Math" alttext="\neg A\in\Delta" display="inline"><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> for any wff <math id="p6.m4" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math>. Suppose <math id="p6.m5" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math> is a consistent superset of <math id="p6.m6" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math>. If <math id="p6.m7" class="ltx_Math" alttext="A\in\Gamma-\Delta" display="inline"><mrow><mi>A</mi><mo>∈</mo><mrow><mi mathvariant="normal">Γ</mi><mo>-</mo><mi mathvariant="normal">Δ</mi></mrow></mrow></math>, then <math id="p6.m8" class="ltx_Math" alttext="\neg A\in\Delta" display="inline"><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow><mo>∈</mo><mi mathvariant="normal">Δ</mi></mrow></math> by assumption, which means <math id="p6.m9" class="ltx_Math" alttext="\neg A\in\Gamma" display="inline"><mrow><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow><mo>∈</mo><mi mathvariant="normal">Γ</mi></mrow></math> since <math id="p6.m10" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math> is a superset of <math id="p6.m11" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math>. But then both <math id="p6.m12" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math> and <math id="p6.m13" class="ltx_Math" alttext="\neg A" display="inline"><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow></math> are <a class="nnexus_concept" href="http://mathworld.wolfram.com/Deducible.html">deducible</a> from <math id="p6.m14" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math>, contradicting the assumption that <math id="p6.m15" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math> is consistent. Therefore, <math id="p6.m16" class="ltx_Math" alttext="\Gamma" display="inline"><mi mathvariant="normal">Γ</mi></math> is not a <a class="nnexus_concept" href="http://mathworld.wolfram.com/ProperSuperset.html">proper superset</a> of <math id="p6.m17" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math>, or <math id="p6.m18" class="ltx_Math" alttext="\Gamma=\Delta" display="inline"><mrow><mi mathvariant="normal">Γ</mi><mo>=</mo><mi mathvariant="normal">Δ</mi></mrow></math>. ∎</p> </div> </div> <div id="p7" class="ltx_para"> <p class="ltx_p"><span class="ltx_text ltx_font_bold">Remarks</span>.</p> <ul id="I4" class="ltx_itemize"> <li id="I4.i1" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_itemize">•</span> <div id="I4.i1.p1" class="ltx_para"> <p class="ltx_p">In the converse of 2, we require that <math id="I4.i1.p1.m1" class="ltx_Math" alttext="\Delta" display="inline"><mi mathvariant="normal">Δ</mi></math> be a theory, for there are complete consistent sets that are not deductively closed. One such an example is the set <math id="I4.i1.p1.m2" class="ltx_Math" alttext="V" display="inline"><mi>V</mi></math> of all <a class="nnexus_concept" href="http://mathworld.wolfram.com/PropositionalVariable.html">propositional variables</a>: it can be shown that for every wff <math id="I4.i1.p1.m3" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math>, exactly one of <math id="I4.i1.p1.m4" class="ltx_Math" alttext="V\vdash A" display="inline"><mrow><mi>V</mi><mo>⊢</mo><mi>A</mi></mrow></math> or <math id="I4.i1.p1.m5" class="ltx_Math" alttext="V\vdash\neg A" display="inline"><mrow><mi>V</mi><mo>⊢</mo><mrow><mi mathvariant="normal">¬</mi><mo>⁢</mo><mi>A</mi></mrow></mrow></math> holds.</p> </div> </li> <li id="I4.i2" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_itemize">•</span> <div id="I4.i2.p1" class="ltx_para"> <p class="ltx_p">So far, none of the above actually tell us that a maximally consistent set exists. However, by Zorn’s lemma, it is not hard to see that such a set does exist. For more detail, see here (<span class="ltx_text ltx_font_typewriter">http://planetmath.org/LindenbaumsLemma</span>).</p> </div> </li> <li id="I4.i3" class="ltx_item" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_itemize">•</span> <div id="I4.i3.p1" class="ltx_para"> <p class="ltx_p">There is also a <a class="nnexus_concept" href="http://planetmath.org/logic">semantic</a> <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">characterization</a><sup style="display: none;"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Characterization.html"><img src="http://mathworld.wolfram.com/favicon_mathworld.png" alt="Mathworld"></img></a><a class="nnexus_concept" href="http://planetmath.org/characterisation"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> of a maximally consistent set: a set is maximally consistent iff there is a unique <a class="nnexus_concepts" href="javascript:void(0)" onclick="this.nextSibling.style.display='inline'">valuation</a><sup style="display: none;"><a class="nnexus_concept" href="http://planetmath.org/valuation"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a><a class="nnexus_concept" href="http://planetmath.org/truthvaluesemanticsforclassicalpropositionallogic"><img src="http://planetmath.org/sites/default/files/fab-favicon.ico" alt="Planetmath"></img></a></sup> <math id="I4.i3.p1.m1" class="ltx_Math" alttext="v" display="inline"><mi>v</mi></math> such that <math id="I4.i3.p1.m2" class="ltx_Math" alttext="v(A)=1" display="inline"><mrow><mrow><mi>v</mi><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>1</mn></mrow></math> for every wff <math id="I4.i3.p1.m3" class="ltx_Math" alttext="A" display="inline"><mi>A</mi></math> in the set (see here (<span class="ltx_text ltx_font_typewriter">http://planetmath.org/CompactnessTheoremForClassicalPropositionalLogic</span>)).</p> </div> </li> </ul> </div> <div id="p8" class="ltx_para ltx_align_right"> <table class="ltx_tabular ltx_guessed_headers ltx_align_middle"> <tbody class="ltx_tbody"> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t">Title</th> <td class="ltx_td ltx_align_left ltx_border_r ltx_border_t">maximally consistent</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l"><a class="nnexus_concept" href="http://planetmath.org/canonical">Canonical</a> name</th> <td class="ltx_td ltx_align_left ltx_border_r">MaximallyConsistent</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Date of creation</th> <td class="ltx_td ltx_align_left ltx_border_r">2013-03-22 19:35:13</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Last modified on</th> <td class="ltx_td ltx_align_left ltx_border_r">2013-03-22 19:35:13</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Owner</th> <td class="ltx_td ltx_align_left ltx_border_r">CWoo (3771)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Last modified by</th> <td class="ltx_td ltx_align_left ltx_border_r">CWoo (3771)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Numerical id</th> <td class="ltx_td ltx_align_left ltx_border_r">10</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Author</th> <td class="ltx_td ltx_align_left ltx_border_r">CWoo (3771)</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Entry type</th> <td class="ltx_td ltx_align_left ltx_border_r">Definition</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l"><a class="nnexus_concept" href="http://mathworld.wolfram.com/Classification.html">Classification</a></th> <td class="ltx_td ltx_align_left ltx_border_r">msc 03B05</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Classification</th> <td class="ltx_td ltx_align_left ltx_border_r">msc 03B10</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Classification</th> <td class="ltx_td ltx_align_left ltx_border_r">msc 03B99</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Classification</th> <td class="ltx_td ltx_align_left ltx_border_r">msc 03B45</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Related topic</th> <td class="ltx_td ltx_align_left ltx_border_r">FirstOrderTheories</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l">Defines</th> <td class="ltx_td ltx_align_left ltx_border_r">complete</td> </tr> <tr class="ltx_tr"> <th class="ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_l ltx_border_t"><span class="ltx_ERROR undefined">\@unrecurse</span></th> <td class="ltx_td ltx_border_t"></td> </tr> </tbody> </table> </div> </article> </div> <footer class="ltx_page_footer"> <div class="ltx_page_logo">Generated on Thu Feb 8 19:23:43 2018 by <a href="http://dlmf.nist.gov/LaTeXML/">LaTeXML <img 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