# Tagged Questions

For questions on axioms, mathematical statements that are accepted as being true, usually without controversy.

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### What does category theory say about chains of set inclusions $\dots\in c\in b\in a$?

I'm currently trying to understand to what extend a set theory like ZFC can be mirrored within category theory, i.e. topos theory. What appears as an obstacle to me is the axiom of regularity, which ...
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### Prove that (M,+,*) is a field

Prove that the multiplication $*:M \times M \to M$ defined by this table: * | 0 1 -------- 0 | 0 0 1 | 0 1 together with the commutative group (M,+), is a field (M,+,*). Group axioms: 1) ...
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### Axioms of Newtonian Mechanics

Axiomatically speaking, could Newton's laws be derived (as theorems) from the conservation of momentum and energy -- along with a few suitable definitions of things like an inertia frame and force? ...
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### Axioms as recreational mathematics

Before modern group theory, mathematicians studied concrete permutation groups: algebraically closed subsets of the set of all bijections on a set $X$ in which all inverses was included. This was the ...
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### Has anyone proposed an axiom for ZFC that implies that many different cardinal characteristics are distinct?

I think it is more interesting when cardinal numbers are distinct, rather than equal. Has anyone proposed an axiom for ZFC that, in one fell swoop, implies that many different cardinal characteristics ...
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### Neighborhood Topology and Open Set Topology: their Equivalence and Comparison

Motivation. A few years ago we were using Armstrong's Basic Topology as a textbook for the topology course in my university, and right off the bat I had a huge conceptual problem regarding the two ...
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### Consequences of the Fact Every Uncountable Set of Reals Has Perfect Set Property

I have recently been working through the consequences assuming the Axiom of Determinacy has and came across the fact that this axiom implies that every uncountable set of reals has the perfect set ...
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### Incompleteness theorems in encoding schemes other than Gödel numbering

Gödel's proof of his incompleteness theorems makes use of Gödel numbering, which is a device that allows a theory of arithmetic $S$ (e.g. PRA) to express and reason about metamathematical statements ...
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### Ordinal Numbers within Simple Type Theory

A little preamble: I recently came across the system of higher-order logic known as simple type theory, and I was immediately intrigued by it, as it seemed to be exactly what I was looking for as an ...
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### is axiom of powers required?

we can form any subset of a set using axiom of specification. then we can form collection of subsets using axiom of pairing repeatedly. we have for every condition S , (A={x belongs to X & S(x)} =...
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### Are there any articles (or otherwise) that discuss the idea that the set-theoretic universe should be as “free” as possible?

Question. Are there any articles (or otherwise) that discuss (and perhaps attempt to formalize) the idea that the set-theoretic universe should be as "free" as possible? Discussion. Philosophically ...
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### Is there an axiom for ZFC that totally orders Cichon's diagram without collapsing it?

Is there a known axiom for ZFC that: Fixes a total ordering on the cardinal numbers appearing in Cichon's diagram. Is "natural looking" - in particular, its not allowed to be a conjunction of ...
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### Is “autonym” an autonym?

I believe this is set theory, but I don't really know set theory, so... An autonym is a word that refers to a property that the word itself happens to possess. For example "noun" is an autonym ...
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### Congruent figures have equal areas. Why?

When we are first introduced to the idea of congruent polygons (generally triangles) in school, we often define congruent figures as "figures which have the same shape and size". However, today, ...
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### Set of true statements generated by set of axioms with a binary operator

I wondered about this and I am having a hard time formulating it as a question at all, but I hope I can express something if my wondering here. Assume we have a set $V = \{\mathbb N, +, =, X ,(,)\}$. ...
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### Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there is ...
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### Is a “model” only a proper model if everything in it's definition is also explicitly constructed?

Say you have some collection of axioms and you find a set $X$ fulfilling them. And the definition of the set $X$ involves another concept, e.g. the real number, but only in a way which refers to the ...
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### Using definitions instead of axioms.

Lets take (classical) first-order logic for granted, including an equality symbol and its associated axioms. Given all this, a rigorous work of mathematics will typically begin with a signature - ...
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### Should every line be infinite in both two directions?

Yesterday one my classmate asked me for the the definition of line. I have found it in coordinate geometry, in vector space, and in metric space. The definition in metric space is quite general ...
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### Scotts axiom, representation theorems for Qualitative -Numerical Probability function relations

Scotts theorem/axiom and other representation theorems give conditions under which a qualitative ordering (>= for at least as probable than) which satisfies certain constraints (total pre-order, ...
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### What kind of Properties are Allowed in the Schema of Comprehension?

Studying an introduction to set theory, we were presented with several axioms as the Axiom Schema of Comprehension and the Axiom of Infinity. This last aximom allows the definition of the inductive ...
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### I'm stuck on a proof involving the of Axiom for multiplicative inverses and modular arithmetic.

I am trying to show that the axiom of multiplicative inverses holds on sets of integers modulo P when P is prime. i just need to show that for any non zero integer, n less than P there is a unique ...
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### Can you characterize dimension in terms of the touching axioms?

One alternative axiomatization of topology is the touching axioms. It's basically the same as the closure axioms, and is in fact equivalent to the open set definition. Can we define dimension in a ...
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### Probability without second axiom (unit measure)

I'm working with functions (namely, representing incoherent degrees of belief) which resemble probabilities, but are actually, say, quasi-probabilites: their values on atomic events (here: atomic ...
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### Peano's 3rd axiom -explain

Peano's axioms for Natural numbers. 3rd axiom from Introduction to topology by Bert Mendelson - There is one and only one object in $\mathbb{N}$ denoted by $1$,...
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### Are my proofs consistent/non-circular/correct?

I am slowly working my way trough "Mathematical Analysis" by Apostol, and I decided to try my hand at a few proofs. (I asked related questions here and here, so some of the work here might be from the ...
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### Power Set, Theory of Linear Algebra, and Axiomatic Systems

So for homework, my History of Math professor gave us these three questions: Explain why the power set of a set S (collection of all subsets) has the same cardinality as the set [all functions from ...
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### Huzita-Hatori, 4th and 5th axiom

https://en.wikipedia.org/wiki/Huzita%E2%80%93Hatori_axioms It seems to me like the 4th axiom is just a special case of the 5th. I'm actually confused, because I can't find the definite form of these ...
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### do the line order axioms imply the plane separation axiom?

Pasch's axiom and plane separation can be shown equivalent. But both depend on the line ordering (or betweenness) axiom for meaning. Does the latter imply those two axioms? I read not, but I seem to ...
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### Write all axioms and properties for the Boolean algebra of sets P(S) (power set)

Write all axioms and properties for the Boolean algebra of sets: $S = set$ $(P(S), \cap$$, ∪ , complement; ∅, S)$ I know the axioms of Boolean algebra but I am not sure how to translate that to a ...
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### A simple question about rational numbers without a simple proof?

As in this question, study the quasigroup $(Q_+,/)$ of positive rational numbers under division. There are two obvious identities: $a/(b/c)=c/(b/a)$, for all $a,b,c\in Q_+$ $(a/b)/c=(a/c)/b$, for ...
### Order Axioms: '<' or '$\leq$'?
In the axioms given for the real numbers, I see that the order axioms are sometimes given for the '<' relation and sometimes for '$\leq$'. Which is more commonly used these days?