All Questions
1,142 questions
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Does Regularity schema imply $\in$-induction when added to first order Zermelo set theory?
That $\in$-induction fails to be a theorem schema of first order Zermelo + Foundation (see here), then it appears that it is more eligible to replace axiom of Regularity (Foundation) by a Regularity ...
- 11.3k
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1
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213
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Is there an effective way to generalize this approach of affinely extending the number line?
The general approach here is a follow up of the approach outlined in a prior posting on extending the projectively extended real line. In particular arithmetic operators break down to ternery ...
- 11.3k
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1
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665
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Can Ackermann set theory find a natural interpretation in light\heavy class dichotomy?
I want to coin the notion of heaviness of a class as a function from classes to ordinals such that $$heaviness(x)=|TC(x)|$$, i.e. the heaviness of a class is the cardinality of its transitive closure.
...
- 11.3k
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708
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What is the consistency strength of this kind of iterating Berkeley cardinals?
[EDIT] After suggestion from Monroe Eskew, and after having an e-mail correspondence with Prof. Joan Bagaria, I'll re-present the older question as the second in a series of 4 questions. I've had an e-...
- 11.3k
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Construction of the smallest nucleus above a prenucleus: what does this proof tell us?
While reading Hyland's paper on the effective topos [retyped version here] in the L. E. J. Brouwer Centenary Symposium, specifically prop. 16.3, I realized that the following proposition is implicit:
...
- 32.5k
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112
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What's the consistency strength of adding this inference rule to Ackermann's set theory?
Working in the language of Ackermann set theory:
Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $\...
- 11.3k
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68
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Computable in $\omega$-REA degree but not double jump of finitely many columns
Suppose that $A$ is an $\omega$-REA set (so $A^{[n]}$ is r.e. in the prior columns). It is a well-known result that if each column of $A$ is computable then $A \leq_T \emptyset^2$ ($\emptyset^2$ can ...
- 3,029
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267
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The "higher topology" of countable Scott sets
Fix some computable bijection $b$ between $\omega$ and $2^{<\omega}$. For $r\in 2^\omega$, let $$[r]=\{f\in 2^\omega: \forall\sigma\prec f(b^{-1}(\sigma)\in r)\}$$ be the closed subset of Cantor ...
- 20.5k
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654
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Completeness of Algebraically Closed Valued Fields(ACVF) Theory
One can prove Elimination of Quantifiers of ACVF finding an extension of any partial embedding of a model $K$ into a $|K|^+$ Saturated one using the language $\mathcal{L} = ( 0,1,+,*, U, \mid )$. In ...
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293
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Compactness for countable models?
How and where is it proved that WKL$_0$ proves the compactness theorem for countable models? (This is a follow-up to a comment of F. Dorais.)
- 16.6k
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893
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Has there been any mathematical study of causality?
Causality seems to play an important role in physics. There also seems to be a close parallel between "$P$ causes $Q$" and "if $P$ then $Q$." Mathematical logic studies logical ...
- 3,094
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213
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Self-embeddings of uncountable total orders
A nice theorem of Dushnik and Miller (from 1940) states that if $(\Omega,\leq)$ is a countable total order, then either
there is an element $\omega \in \Omega$ such that $(\Omega \setminus \{\omega\}...
- 4,547
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96
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If $H$ is commutative and unit-cancellative, then so is the monoid of non-empty ideals of $H$
Let $H$ be a (multiplicatively written) commutative monoid with identity $1_H$. Given $X, Y \subseteq H$, we take
$$XY := \{xy: x \in X,\, y \in Y\}.$$
We call a set $I \subseteq H$ an ideal of $H$ ...
- 10.5k
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347
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Is the following theory countably axiomatizable?
Let $L$ be the language of set theory, and for each countable ordinal $\kappa$, define $L_\kappa$ as follows: $L_0 = L$, and $L_{\alpha+1}$ is obtained by adding countably many new class constants $c_\...
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109
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Is every formula of LΩ equivalent to a formula of L1 modulo T1?
Q: Is every formula of $L_\Omega$ equivalent to a formula of $L_1$ modulo $T_1$?
The notation comes from the following question: Is the following theory countably axiomatizable?
Edit: I mean $T_\...
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1
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171
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How do chains of elementary extensions compare to shrewdness?
I thought of the following large cardinal axiom, extending the notion of $\theta$-upliftingness:
Let $\eta$ be be an ordinal, and $X$ be a class of ordinals. $\kappa$ is called $\eta$-iteratively ...
- 704
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276
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About having one axiom schema for ZFC motivated after the iterative conception of sets?
This posting is related to this posting, and builds its motivation from this answer to it.
Define: $\operatorname {History}(x) \iff \\\forall y \in x: y=\{c \mid \exists z : z \in y \cap x \land (c \...
- 11.3k
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127
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Totally computable least real upper bounds for bounded recursive sets of totally computable real numbers?
A recursive set $Y$ is a set with a characteristic total computable function $\chi_Y$ so that $\chi_Y(n)=0$ iff $n\in Y$ and $\chi_Y(n)=1$ iff $n\notin Y$. Let a $recursive \ condition$ be one which ...
- 3,574
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598
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Can Godel's incompleteness theorems be in some sense circumvented this way?
New foundations "NF" (formulated in the language of $\small \sf FOL(\in)$), can define a kind of ordered pair relation $``\rho"$ such that we can have a set $E$ of those pairs where NF proves the ...
- 11.3k
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148
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Is this reflection schema equivalent to second order Bernays reflection?
This posting comes as a possible salvage to this earlier presented reflective theory which was proved inconsistent. Attesting the particulars of the underlying language in reflection axioms.
Working ...
- 11.3k
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525
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Is acyclic ZF consistent?
I need to know if the following system is consistent, because I want to use it in presenting automorphisms over stratified versions of it.
The system I'd label as "Acyclic ZF", which is $\...
- 11.3k
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1k
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Can ZFC commit cardinality errors?
Add a primitive a one place function symbol $c$ to represent "True cardinality" of a set, to the first order language of set theory.
Add the following axiom schema:
1. Cardinal Equality: If $\phi(x,...
- 11.3k
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403
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When does a power semigroup have a zero, and what can the zero be?
Let $S$ be a semigroup. The power semigroup of $S$ is the set $P(S)=2^S\setminus\lbrace\varnothing\rbrace $ with the operation $$AB=\lbrace ab\ |\ a\in A,\ b\in B\rbrace.$$
This operation is ...
- 1,445
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387
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Is there a known shorter axiomatization of NF than this?
Is there an already known axiomtization of $NF$ that is shorter than the following axiomatic system in first order logic with equality $``="$ and membership $``\in"$? And what is exactly meant by ...
- 11.3k
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463
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Is a function needed here?
This question is related to my question Can we choose an element from a class?.
However, I decided to create a separate question.
Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed ...
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168
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Can we have a bijection between a set and its powerset with the following properties?
This question is related to a question Is this internalization of a bijection between a set and its powerset possible? lately posted to $\cal MO$. Similarly, we add one primitive unary partial ...
- 11.3k
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112
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Can Foundation be captured for some $\mathcal L_{\omega_1, \omega}$ theories?
This posting is a continuation to an earlier one titled "Can Foundation be captured in $\mathcal L_{\omega_1, \omega}$ ?"
It appears that capturing foundation is problematic at every $\...
- 11.3k
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431
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How bad is Coq proving both $T$ and $\lnot T$? [closed]
Question: How bad is Coq proving both $T$ and $\lnot T$? Can it be abused?
Back in 2011 on the coq-club mailing list there was a thread:
Is the Daniel Schepler's inconsistency real?.
In the thread ...
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1
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261
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Is Proper Class Choice equivalent to Global Choice?
Working in "MK-Regularity-Limitation of Size + Replacement for sets", call it the Base theory, let's coin the following axiom:
Axiom of Super-Choice:$$\forall \ relation \ R \ \exists F \...
- 11.3k
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679
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Naturally definable sets of natural numbers (2): Can the circle be broken?
(follow-up to: Naturally definable sets of natural numbers)
Every formula $\Psi(x)$ in the first-order language of Peano arithmetic defines a set of natural numbers. Some of these sets are finite, ...
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1
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402
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Ubiquity beyond infinity, transitive closure and the recursion theorem?
I am considering a Principle of Ubiquity, expressed as follows - for a class theory where precisely the elements are sets - with the aid of set abstracts:
For $\alpha(y,z)$ a first order condition so ...
- 3,574
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827
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Is it possible to derive the rules of set theory as transfers from the pure finite set world, and can we extend this further?
Informally the idea of this question is about whether the rules of set theory can be derived as a transfer of some rules from the hereditarily finite set realm, and whether this transfer principle ...
- 11.3k
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3
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637
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What is the consistency strength of Z+ Accessibility?
Informally the axiom schema of accessibility states that for each unary function $F$ that is definable over the whole universe of discourse "in the language of set theory", like the powerset function $...
- 11.3k
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1
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167
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If we limit matters what ZFC can prove, would that be consistent?
I was thinking about a principle that occurred to me regarding provability in ZFC and truth. The principle outrageously states that: whatever ZFC shows, it is! In other words whatever ZFC can prove ...
- 11.3k
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1
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192
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Can we have consistent theories stating opposing provability statements that are non-standardly coded?
I want to coin a notion of "strong provability", to be defined as:
$S$ is strongly provable in $T$ if and only if there is a Gödel code of its proof in $T$ that is strictly smaller than any ...
- 11.3k
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1
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155
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Does MK prove internally that there are more proper classes than sets?
Is the following provable in MK?
$\not \exists S: \\ \text{ } \\1. \ \ \forall s \in S \exists a,b (s=\langle a,b \rangle) \\ \text{ } \\2. \ \ \forall x (set(x) \to \exists! X (\neg set(X) \land \{...
- 11.3k
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1
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296
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Can this form of reflection be consistent?
Is this form of reflection consistent?
First I'll begin by clarifying the notation I'm using here:
By a quantifier being relativized or bounded it means that the first occurrence of the quantified ...
- 11.3k
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117
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Can stratification be used to internalize functions on models of $\sf Z$?
Suppose $M$ is a model of $\sf Z +\neg AC$ that is externally bijective to an element $k \in M$. Obviously if $j$ denotes such an external bijection, then it cannot be used in Separation within $M$, ...
- 11.3k
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267
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Is Nested Selection equivalent to AC?
Nested Selection: For every infinite set $G$ of pairwise disjoint infinite sets such that any two distinct elements $x,y$ of $G$ either "$y$ is a set of proper supersets of elements of $x$ and ...
- 11.3k
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1
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198
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Is Bounding Reflection consistent?
Working in the first order language of set theory.
Let $\varphi^{*B}$ be the formula obtained from $\varphi$ by merely bounding all open quantifiers in $\varphi$ by the symbol "$B$".
Here a ...
- 11.3k
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3
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2k
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Infinite CPU clock rate and hotel Hilbert [closed]
Suppose we have a computer with infinte CPU clock rate, infinite CPU registers, storage etc. Lets run a program that could look something like this:
A=1
while A>0
A = A+1
repeat
We start the program ...
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1
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351
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Are there overwhelmingly more finite monoids than finite spaces? [closed]
A function $f:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ overwhelms $g:\mathbb{Z}_{\geq 1}\to\mathbb{Z}_{\geq 1}$ if for any $k\in \mathbb{Z}_{\geq 1}$ the inequality $f(n)\leq g(n+k)$ holds only for ...
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