Questions tagged [axioms]
The axioms tag has no usage guidance.
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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 ...
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Can existence of uncountable sets be proved in a ZFC variant with mild definability restriction?
Starting with ZF[C], if we restrict Separation and Replacement to parameter free versions from Parameter free definable sets, re-write Infinity asserting existence of $\omega$. Add to it axioms of ...
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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 ...
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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 ...
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Axiomatic definition of integers
The real numbers can be axiomatically defined (up to isomorphism) as a Dedekind-complete ordered field.
What is a similar standard axiomatic definition of the integer numbers?
A commutative ordered ...
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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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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-...
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Counterintuitive consequences of the Axiom of Determinacy?
I just read Dr Strangechoice's explanation that if all subsets of the real numbers are Lebesgue measurable, then you can partition $2^\omega$ into more than $2^\omega$ many pairwise disjoint nonempty ...
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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....
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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 ...
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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 ...
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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 ...
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What are the advantages of the more abstract approaches to nonstandard analysis?
This question does not concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the ...
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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 ...
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How would calculus be possible in a finitist axiom system?
I am interested in learning a little more about finitism, currently about which I only know a few encyclopedic paragraphs.
I know that during some time, some mathematicians like Kronecker thought ...
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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 $\{X_n,Y_n,Z_n,\dots\}_{n<\omega}$ ...
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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 ...
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Axiomatic approach to means
Recently I have been contemplating on a talk for high school children. One of my favorite topics in high school was the inequality of means. I had a great high school teacher who wrote some very nice ...
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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 ${\equiv}\subseteq X^2\times X^2$ ...
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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(...
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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 ...
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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{-...
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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$
...
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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 ...
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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 $\...
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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 ...
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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, ...
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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 \...
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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. ...
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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
$$
G\...
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Are there re-formulations of ZFC that more closely parallel large-cardinal extensions?
The following is a reformulation of $\sf ZFC$, it is a version of a well known approach going back to Dana Scott (I suppose), it axiomatizes $\sf ZFC$ simply by "Specification, Reflection, and ...
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Is there a complete countable axiomatization of conditional independence? (Graphoids)
Note: A pointer to a reference, or a yes/no answer with a 1-2 sentence incomplete/non-rigorous justification would suffice for answers. I am just curious about whether the result is true; it is fairly ...
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What is the proof of replacement from Bernays first order reflection?
In Kanamori's Bernays and Set Theory pages 20-21, a first order reflection principle due to Bernays is mentioned, that of:
$$\sf \varphi \to \exists y \, (\text {Trans}(y) \land \varphi^y)$$ for ...
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Is choice over definable sets equivalent to AC over axioms of ZF-Reg.?
If we add the following axiom schema to ZF-Reg., would the resulting theory prove $\sf AC$?
Definable sets Choice: if $\phi$ is a formula in which only the symbol $``y"$ occurs free, then:
$$\forall X ...
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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 \\\...
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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 \...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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: $\forall z (z \in x \leftrightarrow z \in y) \to x = y$
...
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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.
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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 ...
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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 ...
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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 ...
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Consistency of: "The continuum function is injective, and for all infinite cardinals $\kappa$ we have that $2^\kappa$ is weakly inaccessible."
I asked here about "large powerset axioms" and to my delight, learned that such axioms are being taken seriously. I've been toying with them ever since. My favourite is: "The continuum function is ...
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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 ...
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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 ...