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Are there known examples like this almost official exposition of ZFC that is very weak?

Pseudo-ZFC is a theory written in the usual language of set theory, i.e. mono-sorted first order logic with equality and membership. The extra-logical axioms are: Extensionality: $\forall x \forall y:...
Zuhair Al-Johar's user avatar
14 votes
1 answer
1k views

Hilbert's sixth problem and QFT description

The Wikipedia entry on Hilbert's sixth problem about QFT description is “Since the 1960s, following the work of Arthur Wightman and Rudolf Haag, modern quantum field theory can also be considered ...
XL _At_Here_There's user avatar
10 votes
1 answer
241 views

How short can the axioms of propositional logic be?

There are a number of axiom systems for classical propositional calculus. Here, I focus on those which use negation ($\neg$) and implication ($\to$) as the connectives, with Modus Ponens and ...
Joel's user avatar
  • 103
4 votes
1 answer
255 views

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 . \...
user76284's user avatar
  • 1,793
3 votes
0 answers
111 views

Is it consistent to have an infinite antitone sequence of elementary embeddings such that the involved models include iterated sharps?

$\DeclareMathOperator\crit{crit}$Background essays (the material I've tried to understand in leading up to this question): Daghighi, et. al. [2014], "The foundation axiom and elementary self-...
Kristian Berry's user avatar
7 votes
1 answer
229 views

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....
parallelogram's user avatar
6 votes
1 answer
524 views

Does Playfair imply Proclus?

I am interested in the interplay between the Playfair and Proclus Axioms in linear spaces. By a linear space I understand a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of ...
Taras Banakh's user avatar
  • 40.7k
4 votes
1 answer
231 views

How to use Meredith’s axiom for classical logic?

I’ve been self-studying axiomatic systems for classical logic for a while. The standard Hilbert/Mendelssohn/Lukasiewicz axiomatizations were a bit tough for me to get used to without using the ...
PW_246's user avatar
  • 184
6 votes
4 answers
609 views

Axiomatic system made just for playing

The formalization of mathematics is based on axioms and theorems logically concluded from them. This way we construct solid structures to model different areas of the human knowledge: different branch ...
A. J. Pan-Collantes's user avatar
2 votes
0 answers
256 views

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 ...
user76284's user avatar
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6 votes
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146 views

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 ${\equiv}\subseteq X^2\times X^2$ ...
Taras Banakh's user avatar
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51 votes
7 answers
7k views

Does anyone still seriously doubt the consistency of $ZFC$?

As someone self-taught in set theory beginning with Donald Monk’s excellent book on MK set theory, $ZFC$ has always seemed like a weak set theory. Despite this, the majority of professional ...
6 votes
1 answer
227 views

Strengthening Quine's New Foundations with a more flexible stratification criterion?

Let's say that a formula in the language of set theory is flexibly stratified iff there exists a function $f$ from variable symbols to $\omega$ such that if $x=y$ appears in the formula, then $f(x)=f(...
Dylan Pizzo's user avatar
4 votes
1 answer
221 views

Is the affine geometry a geometry of proportions?

Given any linear space $L$ over an ordered field $F$, consider the equiproportion relation $${\sim}=\{((x,y,z),(a,b,c))\in L^3\times L^3: \exists t\in[0,1]\subseteq F\;(y{-}x=t(z{-}x)\wedge b{-}a=t(c{-...
Taras Banakh's user avatar
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9 votes
3 answers
1k views

Axioms for the category of groups

Certain categories of mathematical structures have had synthetic axiom systems developed for them. One particularly well known such category is the category of sets and functions $\mathit{Set}$, which ...
Madeleine Birchfield's user avatar
5 votes
1 answer
165 views

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: $\phi \to \exists u \phi^u$ ...
user76284's user avatar
  • 1,793
10 votes
0 answers
209 views

Is any choice axiom other than WISC inherited by Grothendieck topoi?

It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one ...
saolof's user avatar
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1 vote
0 answers
117 views

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 $\...
Zuhair Al-Johar's user avatar
7 votes
2 answers
576 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 $\{X_n,Y_n,Z_n,\dots\}_{n<\omega}$ ...
Alec Rhea's user avatar
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4 votes
1 answer
179 views

Predicativity and axiom $\mathbb{R}\flat$ in real cohesive homotopy type theory

In Mike Shulman's article Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, the fundamental axiom adopted for his real-cohesive homotopy type theory (axiom $\mathbb{R}\flat$), which ...
Madeleine Birchfield's user avatar
1 vote
0 answers
102 views

What is the set of axioms for this theory extracted from what is provable in the minimal model of ZFC?

This post is a follow up of this one posted to MathStackExchange. Working in $\sf ZFC + \exists M \, (M \overset{trs} \models ZFC)$, lets define a theory $T_0$ as: $$(\varphi \ \epsilon \ T_0) \iff \...
Zuhair Al-Johar's user avatar
0 votes
1 answer
134 views

Reflection schema

Peano Arithmetic consists of axioms $P_1, P_2, \ldots P_7$ plus first order classical logic. Let us call this theory $T$. This theory has its unprovable Gödel’s sentence $G$ such that $$ G\...
Newberry's user avatar
2 votes
0 answers
200 views

Are there re-formulations of ZFC that more closely parallel large-cardinal extensions?

The following is a reformulation of $\sf ZFC$, it is a version of a well known approach going back to Dana Scott (I suppose), it axiomatizes $\sf ZFC$ simply by "Specification, Reflection, and ...
Zuhair Al-Johar's user avatar
4 votes
1 answer
327 views

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, ...
user avatar
0 votes
1 answer
237 views

How large is the supremum of minimal $V$-heights of all first-order set theories formulated in a particular language of FOST?

Fix a language $\mathcal{L}$ of first-order set theory. For this question, we can assume that $\mathcal{L}$ is the language described in Chapter 1 of “An introduction to set theory” [William A. R. ...
lyrically wicked's user avatar
11 votes
1 answer
496 views

Is every set being cardinal definable consistent with ZF + negation of Choice?

Recall the definition of cardinal definable, where every set being cardinal definable is proved consistent relative to ZF + V=HOD. To re-iterate it: $Define: X \text { is cardinal definable} \iff \\\...
Zuhair Al-Johar's user avatar
2 votes
0 answers
81 views

Is restricting class parameters to be arguments of set functions in reflection consistent?

Working in mono-sorted first order logic with equality and membership: Define: $\operatorname{set}(x) \equiv_\text{df} \exists y \, (x \in y)$ Axiomatize: Extensionality: $(\forall x \, (x \in a \...
Zuhair Al-Johar's user avatar
0 votes
1 answer
136 views

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 ...
Zuhair Al-Johar's user avatar
0 votes
1 answer
271 views

What is the proof of replacement from Bernays first order reflection?

In Kanamori's Bernays and Set Theory pages 20-21, a first order reflection principle due to Bernays is mentioned, that of: $$\sf \varphi \to \exists y \, (\text {Trans}(y) \land \varphi^y)$$ for ...
Zuhair Al-Johar's user avatar
10 votes
1 answer
302 views

Is choice over definable sets equivalent to AC over axioms of ZF-Reg.?

If we add the following axiom schema to ZF-Reg., would the resulting theory prove $\sf AC$? Definable sets Choice: if $\phi$ is a formula in which only the symbol $``y"$ occurs free, then: $$\forall X ...
Zuhair Al-Johar's user avatar
0 votes
1 answer
393 views

Is this theory equivalent to MK?

[EDIT] The older exposition of this theory was proved inconsistent by EmilJeřábek (see comments). Here, this is a possible salvage. (the new information over the older post shall be put in square ...
Zuhair Al-Johar's user avatar
0 votes
0 answers
162 views

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 ...
Zuhair Al-Johar's user avatar
2 votes
1 answer
257 views

Can we internalize a bijection between a set and its powerset in this way?

This question is related to a question lately posted to $\cal MO$. Here, we add two partial unary functions $``j,f"$ to the language of $\sf ZF$. The question is about if we can add the following on ...
Zuhair Al-Johar's user avatar
12 votes
0 answers
530 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 ...
user76284's user avatar
  • 1,793
1 vote
0 answers
186 views

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: $\forall z (z \in x \leftrightarrow z \in y) \to x = y$ ...
user76284's user avatar
  • 1,793
4 votes
2 answers
515 views

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. ...
Yes's user avatar
  • 105
2 votes
0 answers
192 views

What is the strength of the single Replacement sentence?

What's the consistency strength of adding the following single sentence replacement like statement to $\sf Z + \forall x \exists \alpha: x \in V_\alpha$ ? $$\forall \varphi \forall A \ [\forall x \in ...
Zuhair Al-Johar's user avatar
8 votes
1 answer
1k views

Was there a time in mathematics when a counterexample was wrong? [closed]

I am doing an essay on the knowledge of Mathematics and how we know what we know to be true. I was just wondering if there was an example in mathematics of some theorem that was disproven by a ...
Mage Magic's user avatar
1 vote
0 answers
254 views

Is this theory using defined notions of classes, sets, and membership interpretable in ZFC?

The main difference with this formal theory is that it depends in an essential manner on a defined notion of class, set, and set membership $\in$, rather than the usual appraoch of leaving them ...
Zuhair Al-Johar's user avatar
2 votes
0 answers
311 views

Can we have $\sf V=HTD$? How it relates to $\sf V=HOD$?

In my search for some type-set motivated line of thought that might prove axiom of choice, I was thinking of a concept that looks like $\sf V=HOD$, but in terms of types instead of ordinals, that is ...
Zuhair Al-Johar's user avatar
2 votes
2 answers
427 views

Can Z + Ranks + Successor cardinals + Ordinal inaccessibility be equal to ZF?

[EDIT: The axiom of successor cardinals was found by an answer by Greg Kirmayer, not to be capturing the intended meaning of it, which is simply reflected by its name, i.e. the existence of a ...
Zuhair Al-Johar's user avatar
12 votes
3 answers
1k views

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, ...
user avatar
26 votes
3 answers
3k views

What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?

First, some context. Ever since I was a high schooler, I have been fascinated with large numbers. As I have grown in mathematical maturity, I have become both disappointed and fascinated to see that ...
exfret's user avatar
  • 479
11 votes
1 answer
1k views

Is this set theory equivalent to ZFC?

Consider a variant of set theory with these axioms: Extensionality, Regularity (foundation), Separation, Powerset, Axiom of Choice, and Transitive closure of a set-like relation is set-like. Update: ...
Vladimir Reshetnikov's user avatar
1 vote
0 answers
487 views

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 ...
Tangent's user avatar
  • 11
10 votes
1 answer
730 views

When does a topos satisfy the axiom of regularity?

In categorical set theory, we observe that certain topoi satisfy (suitable versions of) certain axioms from set theory. For example, Lawvere's $\mathsf{ETCS}$ asserts that $\mathbf{Set}$ is a well-...
Jordan Mitchell Barrett's user avatar
3 votes
1 answer
442 views

Is axiom of constructibility $V = L$ consistent with Tarski–Grothendieck set theory?

I wonder what is the relationship between ZF + $V = L$ and Tarski–Grothendieck set theory, because I haven't found any bibliographic references. If they are compatible, it is possible to introduce $V =...
user4534237's user avatar
11 votes
2 answers
508 views

Does bounded Zermelo construct any cumulative hierarchy?

ZF is sufficient to construct the von Neumann hierarchy, and prove that every set appears at some stage $V_\alpha$. This is the basis for Scott's trick, for instance. But how much of ZF is needed? Is ...
David Roberts's user avatar
  • 33.4k
3 votes
2 answers
677 views

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 ...
user76284's user avatar
  • 1,793
9 votes
0 answers
249 views

$\Sigma^2_2$ absoluteness and $\diamondsuit$

This question touches on recent questions concerning the role of $\diamondsuit$ and the Continuum Hypothesis. In particular, I would like to know more information about a conjecture of Woodin on the ...
Todd Eisworth's user avatar