All Questions
Tagged with foundations axioms
25 questions
1
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0
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73
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How strong is separation + reflection without transitivity?
Consider a theory $T$ with a binary relation $\in$ and the following axiom schemas:
$\exists u \forall x (x \in u \leftrightarrow x \in a \land \phi)$ where $u$ is not free in $\phi$. This is the ...
- 2,203
14
votes
0
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386
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Can the axiom of choice be expressed in 4 quantifiers?
This 2007 paper presents a 5-quantifier $(\in, =)$-expression that is ZF-equivalent to the axiom of choice, but leaves open the 4-quantifier case:
Thus the gap is reduced to the undecided case of a 4 ...
- 2,203
12
votes
1
answer
225
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Is there a $\Pi_2$ sentence $A$ such that $\text{ZFC}^- + A$ proves powerset?
This is a follow-up to this question.
Let $\text{ZFC}^-$ be ZFC without powerset and with collection rather than replacement, as described here.
Is there a $\Pi_2$ (or perhaps $\Sigma_2$) sentence $A$ ...
- 2,203
5
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3
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664
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Negating fundamental axioms
It is commonplace to consider standard axiomatic systems (e.g. $ZF$) with one of the 'less essential' axioms negated, like infinity, 'less essential' here having some ambiguous definition related to ...
- 10.1k
4
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1
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600
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Why not $\sf ZFC+[V=HOD]$?
Why not $\sf ZFC+[V=HOD]$ as the standard set theory?
It implies the existence of a definable global choice and well-order, and it is compatible with all large cardinal axioms extending $\sf ZFC$, so ...
- 11.3k
4
votes
1
answer
368
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Bounded alternatives to powerset that interpret ZFC
In set theory, many properties/relations of interest can be expressed as $\Delta_0$ formulas (formulas with only bounded quantifiers):
\begin{align}
\text{empty}(a) &\equiv \forall x \in a . \...
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7
votes
1
answer
269
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How did Szmielew prove that Pasch's axiom is a consequence of the circle axiom?
It is alleged that Szmielew proved that Pasch's axiom is a consequence of the circle axiom. The source is said to be
The Pasch axiom as a consequence of the circle axiom, Bull.Acad.Polon.Sci.Sér.Sci....
3
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1
answer
510
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Harvey Friedman: The expanding mind
In reference 1, Friedman writes:
I discuss my efforts concerning 3 crucial issues in the foundations of mathematics that are deeply connected with the great work of Kurt Gödel.
[...]
B. Are there ...
- 2,203
6
votes
0
answers
153
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Does every Tarski plane embed into a 3-dimensional Tarski space?
By a Tarski space I understand a mathematical structure $(X,B,\equiv)$ consisting of set $X$, a betweenness relation $B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ ...
- 41.8k
6
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1
answer
204
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How strong is separation + reflection of unbounded quantifiers?
Consider a set theory with the following axioms:
separation: $\exists y \forall x (x \in y \leftrightarrow \phi \land x \in a)$ where $y$ is not free in $\phi$
reflection: $\phi \to \exists u \phi^u$
...
- 2,203
1
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0
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123
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Is $\sf \Gamma_0$ the proof theoretic ordinal of this kind of predicative class theory?
Adopting the approach of Mono-sorted $\sf NBG$, define sets as elements of classes, then axiomatize:
Extensionality, Predicative Class comprehension, emptyset, in the usual manner along mono-sorted $\...
- 11.3k
7
votes
2
answers
588
views
Consistency strength of an attempt at higher order set theory
Work in a theory with (deep breath) a countable number of primitives denoted with capital letters from the end of the alphabet with numerical subscripts $\{X_n,Y_n,Z_n,\dots\}_{n<\omega}$ ...
- 10.1k
4
votes
1
answer
361
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Consistency of Generalised Continuum Hypothesis and univalence in HoTT
In homotopy type theory, propositional excluded middle and the axiom of choice sets are both consistent with univalence, both of which yields type theoretic models for classical mathematics. However, ...
user173426
12
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0
answers
573
views
Harvey Friedman's minimalist axioms for set theory
[This is a question on the FOM mailing list.]
In 1997, Harvey Friedman introduced the following theory: Let $\in$ be a binary predicate and $U$ be a constant. Add the following axioms:
Subworld ...
- 2,203
1
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0
answers
192
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Does foundationless Ackermann set theory prove replacement?
From Ackermann's set theory equals ZF (1970) by William N. Reinhardt:
Let A be the theory determined by the following axioms:
Extensionality: $\forall z (z \in x \leftrightarrow z \in y) \to x = y$
...
- 2,203
5
votes
2
answers
629
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Applications of ZFA-Set Theory
The set theory with atoms (ZFA), is a modified version of set theory, and is characterized by the fact that it admits objects other than sets, atoms. Atoms are objects which do not have any elements.
...
- 115
13
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3
answers
1k
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Elementary theory of the category of groupoids?
One axiomatisation of set theory, the Elementary Theory of the Category of Sets, or ETCS for short, comes from category theory and states that sets and functions form a locally cartesian-closed, ...
user173426
1
vote
0
answers
522
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Is this a good way of conceptualising the current status of Foundation of Maths projects?
I have been reading 'A Very Short Introduction to Mathematics' by Timothy Gowers and at one point he mentions that most of the mathematical proofs can be finally resolved to a set of logical ...
- 11
3
votes
2
answers
718
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Shortest axiom of infinity for foundationless set theory
Let $T$ be the theory with a binary symbol $\in$, an unary symbol $S$, and the following axioms:
Axiom of extension:
\begin{equation}
\forall x \forall y (\forall z (z \in x \leftrightarrow z \in ...
- 2,203
5
votes
0
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219
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Are any formal systems based upon the idea of "iterated characterization pushing" currently in existence? If not, is anyone working on them?
I had an idea in regards to the design of formal systems with foundational aspirations.
To convey the idea, let's talk a bit about the second-order Peano axioms. The way these axioms work, we have a ...
- 3,793
55
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10
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11k
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How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
Note that "a working mathematician" is probably not the best choice of words, it's supposed to mean "someone who needs the theory for applications rather than for its own sake". Think about it as a ...
- 1,115
-3
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1
answer
262
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An axiomatic system with a set of constants that form a complete ordered field [closed]
I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these ...
- 201
5
votes
1
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191
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Class theory with support for self-application of class functions?
To every natural number $n$, we can assign its Church numeral $\underline{n}.$ A formal definition would be:
$\underline{0}(f)=\mathrm{id}_{\mathrm{dom}(f)}$
$\underline{n+1}(f) = \underline{n}...
- 3,793
4
votes
2
answers
452
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On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$
The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an ...
user36136
15
votes
5
answers
6k
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getting rid of existential quantifiers
It seems to me that for most of the twentieth century, axiomatic foundations for mathematical theories were constructed with the (mostly allied) goals of minimizing the number of primitive notions and ...
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