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Abbe sine condition - Revision history
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:04, 31 July 2013</td>
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text-decoration: none;">[[Richard Platek]]. </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">KP is weaker than [[Zermelo–Fraenkel set theory]] (ZFC). Unlike ZFC, KP does not include the [[power set axiom]], and KP includes only limited forms of the [[axiom of separation]] and [[axiom of replacement]] from ZFC. These restrictions on the axioms of KP lead to close connections between KP, [[Mathematical logic#Recursion theory|generalized recursion theory]], and the theory of [[admissible ordinal]]s.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== The axioms of KP ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Axiom of extensionality]]: Two sets are </ins>the <ins style="font-weight: bold; text-decoration: none;">same if and only if they </ins>have <ins style="font-weight: bold; text-decoration: none;">the same elements.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Epsilon-induction|Axiom of induction]]: If φ(''a'') is </ins>a <ins style="font-weight: bold; text-decoration: none;">[[well-formed formula#Predicate logic|formula]], </ins>and <ins style="font-weight: bold; text-decoration: none;">if for all sets ''x'' it follows </ins>from <ins style="font-weight: bold; text-decoration: none;">the fact that φ(''y'') is true </ins>for <ins style="font-weight: bold; text-decoration: none;">all elements ''y'' of ''x'' that φ(''x'') holds, then φ(''x'') holds for all sets ''x''</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Axiom </ins>of <ins style="font-weight: bold; text-decoration: none;">empty set]]: There exists a set with no members</ins>, <ins style="font-weight: bold; text-decoration: none;">called the [[empty set]] </ins>and <ins style="font-weight: bold; text-decoration: none;">denoted {}. (Note: the existence of a member in the universe of discourse, i.e., ∃x(x=x)</ins>, <ins style="font-weight: bold; text-decoration: none;">is implied in certain formulations</ins><<ins style="font-weight: bold; text-decoration: none;">ref</ins>><ins style="font-weight: bold; text-decoration: none;">{{cite book |title=A course in model theory: an introduction </ins>to <ins style="font-weight: bold; text-decoration: none;">contemporary mathematical logic |last=Poizat |first=Bruno |year=2000 |publisher=Springer |isbn=0-387-98655-3}}, note </ins>at <ins style="font-weight: bold; text-decoration: none;">end of §2</ins>.<ins style="font-weight: bold; text-decoration: none;">3 on page 27</ins>: <ins style="font-weight: bold; text-decoration: none;">&ldquo;Those who do not allow relations on an empty universe consider (∃x)x=x </ins>and <ins style="font-weight: bold; text-decoration: none;">its consequences </ins>as <ins style="font-weight: bold; text-decoration: none;">theses; we, however, do not share this abhorrence, with so little logical ground, of </ins>a <ins style="font-weight: bold; text-decoration: none;">vacuum</ins>.<ins style="font-weight: bold; text-decoration: none;">&rdquo;</ref> of first-order logic, in which case the axiom of empty set follows from the axiom of separation, and is thus redundant</ins>.<ins style="font-weight: bold; text-decoration: none;">)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Axiom of pairing]]</ins>: If <ins style="font-weight: bold; text-decoration: none;">''x'', ''y'' are sets, then so is {''x'', ''y''}, </ins>a <ins style="font-weight: bold; text-decoration: none;">set containing ''x'' and ''y'' as its only elements.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Axiom of union]]: For any set ''x''</ins>, <ins style="font-weight: bold; text-decoration: none;">there is a set ''y'' such that the elements of ''y'' are precisely the elements of </ins>the <ins style="font-weight: bold; text-decoration: none;">elements </ins>of <ins style="font-weight: bold; text-decoration: none;">''x''.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [</ins>[<ins style="font-weight: bold; text-decoration: none;">Axiom schema of predicative separation|Axiom of Σ<sub>0</sub>-separation]]</ins>: <ins style="font-weight: bold; text-decoration: none;">Given any set and any Σ<sub>0</sub>-formula φ(''x''), there is a [[subset]] of the original set containing precisely those elements ''x'' for which φ(''x'') holds. (This is an axiom schema.)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Axiom of collection|Axiom of Σ<sub>0<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>-collection]]: Given any Σ<sub>0<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>-formula φ(''x'', ''y''), if for every set ''x'' there exists a set ''y'' such that φ(''x'', ''y'') holds, then for all sets ''u'' there exists a set ''v'' such that for every ''x'' in ''u'' there is a ''y'' in ''v'' such that φ(''x'', ''y'') holds</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Here, a Σ<sub>0</sub>, or Π<sub>0<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>, or Δ<sub>0<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> formula is one all of whose quantifiers are [[bounded quantifier|bounded]</ins>]. <ins style="font-weight: bold; text-decoration: none;">This means any quantification is the form <math>\forall u \in v</math> or <math>\exist u \in v.</math> (More generally, we would say that a formula is Σ</ins><<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">''n''+1</sub> when it is obtained by adding existential quantifiers in front of a Π<sub>''n''</sub> formula, </ins>and <ins style="font-weight: bold; text-decoration: none;">that it is Π<sub>''n''+1<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> when it is obtained by adding universal quantifiers in front of a Σ<sub>''n''<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> formula: this is related to the [[arithmetical hierarchy]] but in the context of set theory</ins>.<ins style="font-weight: bold; text-decoration: none;">)</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">These axioms differ from ZFC in as much as they exclude the axioms of: infinity, powerset, and choice</ins>. <ins style="font-weight: bold; text-decoration: none;"> Also the axioms of separation and collection here are weaker than the corresponding axioms in ZFC because the formulas φ used in these are limited to bounded quantifiers only</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The axiom of induction in KP is stronger than the usual [[axiom of regularity]] (which amounts to applying induction to the complement of a set (the class of all sets not in the given set))</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Proof that Cartesian products exist ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''Theorem''': If ''A'' and ''B'' are sets, then there is a set ''A''&times;''B'' which consists of all [[ordered pair</ins>]<ins style="font-weight: bold; text-decoration: none;">]s (''a'', ''b''</ins>) <ins style="font-weight: bold; text-decoration: none;">of elements ''a'' of ''A'' </ins>and <ins style="font-weight: bold; text-decoration: none;">''b'' of ''B''.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">'''Proof'''</ins>: <ins style="font-weight: bold; text-decoration: none;">{''a''} = {''a'', ''a''} exists by the axiom of pairing</ins>. <ins style="font-weight: bold; text-decoration: none;">{''a'', ''b''} exists by the axiom of pairing</ins>. <ins style="font-weight: bold; text-decoration: none;">Thus (''a'', ''b'') = { {''a''}, {''a'', ''b''} } exists by the axiom of pairing.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If ''p'' is intended to stand for (''a'', ''b''), then a Δ<sub>0<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> formula expressing that is:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math>\exist r \in p (a \in r \and \forall x \in r (x = a)) \and \exist s \in p (a \in s \and b \in s \and \forall x \in s (x = a \or x = b))<</ins>/<ins style="font-weight: bold; text-decoration: none;">math> and <math>\forall t \in </ins>p <ins style="font-weight: bold; text-decoration: none;">((a \in t \and \forall x \in t (x </ins>= <ins style="font-weight: bold; text-decoration: none;">a)) \or (a \in t \and b \in t \and \forall x \in t (x = a \or x = b))).</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Thus a superset of ''A''</ins>&<ins style="font-weight: bold; text-decoration: none;">times;{''b''} </ins>= <ins style="font-weight: bold; text-decoration: none;">{(''a'', ''b'') | ''a'' in ''A''} exists by the axiom of collection.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Abbreviate the formula above by <math>\psi (a, b, p)\</ins>!<ins style="font-weight: bold; text-decoration: none;">.</math> Then <math>\exist a \in A \psi (a, b, p)</math> </ins>is <ins style="font-weight: bold; text-decoration: none;">Δ<sub>0</sub>. Thus ''A''&times;{''b''} itself exists by the axiom of separation</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If ''v'' is intended </ins>to <ins style="font-weight: bold; text-decoration: none;">stand for ''A''&times;{''b''}, then a Δ<sub>0</sub> formula expressing </ins>that <ins style="font-weight: bold; text-decoration: none;">is:</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math>\forall a \in A \exist p \in v \psi (a, b, p) \and \forall p \in v \exist a \in A \psi (a, b, p).</math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Thus </ins>a <ins style="font-weight: bold; text-decoration: none;">superset of {''A''&times;{''b''} | ''b'' in ''B''} exists by </ins>the <ins style="font-weight: bold; text-decoration: none;">axiom of collection</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Putting <math>\exist b \in B</math> in front of that last formula </ins>and <ins style="font-weight: bold; text-decoration: none;">we get from the axiom of separation that the set {''A''&times;{''b''} | ''b'' in ''B''} itself exists.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Finally, ''A''&times;''B'' = <math>\cup</math>{''A''&times;{''b''} | ''b'' in ''B''} exists by the axiom of union. QED</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Admissible sets ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">A set </ins><<ins style="font-weight: bold; text-decoration: none;">math</ins>> <ins style="font-weight: bold; text-decoration: none;">A\, </ins><<ins style="font-weight: bold; text-decoration: none;">/math</ins>> <ins style="font-weight: bold; text-decoration: none;">is called [[admissible set|admissible]] if it is [[transitive set|transitive]] and <math>\langle A,\in \rangle</math> is a [[model theory|model]] of Kripke–Platek set theory.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">An [[ordinal number]] α is called an [[admissible ordinal]] if [</ins>[<ins style="font-weight: bold; text-decoration: none;">constructible universe|L<sub>α</sub>]] is an admissible set.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The ordinal α is an admissible ordinal if and only if α is a limit ordinal and there does not exist a γ<α for which there is a Σ<sub>1</sub>(L<sub>α</sub>) mapping from γ onto α. If M is a standard model of KP, then the set of ordinals in M is an admissible ordinal.</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">If L<sub>α<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub> is a standard model of KP set theory without the axiom of Σ<sub>0<</ins>/<ins style="font-weight: bold; text-decoration: none;">sub>-collection, then it is said to be an "'''amenable set'''"</ins>.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== See also ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Constructible universe]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Admissible ordinal]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [[Kripke–Platek set theory with urelements]]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== References ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">*{{Cite journal|last= Gostanian|first= Richard|year= 1980|title=Constructible Models of Subsystems of ZF|journal=Journal of Symbolic Logic|volume= 45|issue=2|pages= 237|doi= 10</ins>.<ins style="font-weight: bold; text-decoration: none;">2307/2273185|jstor= 2273185|publisher= Association </ins>for <ins style="font-weight: bold; text-decoration: none;">Symbolic Logic|postscript= <!--None--> }}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><references /></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">{{DEFAULTSORT:Kripke-Platek set theory}}</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Systems of set theory]</ins>]</div></td></tr>
</table>
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https://en.formulasearchengine.com/index.php?title=Abbe_sine_condition&diff=236474&oldid=prev
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