For questions on axioms, mathematical statements that are accepted as being true without a mathematical proof.

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3
votes
2answers
248 views

Why is matrix multiplication defined a certain way? [duplicate]

Why is it that when multiplying a (1x3) by (3x1) matrix, you get a (1x1) matrix, but when multiplying a (3x1) matrix by a (1x3) matrix, you get a (3x3) matrix? Why is matrix multiplication defined ...
7
votes
0answers
161 views

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 ...
11
votes
2answers
540 views

Which axioms of ZFC or PA are known to not be derivable from the others?

Which, if any, axioms of ZFC are known to not be derivable from the other axioms? Which, if any, axioms of PA are known to not be derivable from the other axioms?
6
votes
1answer
137 views

An Exercise in Kunen (A Model for Foundation, Pairing,…)

This is exercise I.4.18 in Kunen's Set Theory. Derive $\forall y (y \notin y)$ from the Axioms of Comprehension and Foundation. Don't use the Pairing or Extensionality Axioms. Then find a 2 element ...
2
votes
1answer
280 views

What's the differences between naive and axiomatic set theory?

I'm at a loss studying math.Recently I decided to begin with set theory as it seems the most fundamental for math.I found the book Naive Set Theory by Halmos,and began to read it because it's so thin ...
2
votes
1answer
226 views

Axioms and deduction rules of Propositional Calculus

I'm looking for a list of axioms and fundamental deduction rules of Propositional Calculus.
24
votes
6answers
1k views

Is $\{0\}$ a field?

Consider the set $F$ consisting of the single element $I$. Define addition and multiplication such that $I+I=I$ and $I \times I=I$ . This ring satisfies the field axioms: Closure under addition. ...
7
votes
3answers
227 views

Axiom of extensionality in ZF - pointless?

In every textbook on ZF set theory I come across the Axiom of extensionality, which basically says that if two sets have the same elements, then those two sets are equal: $$\forall x \forall y ...
2
votes
1answer
126 views

Pythagorean theorem proof by dissection

I have this proof of the Pythagorean theorem, but in the last two lines of the fourth paragraph I can't seem to find geometrically how the congruency between $1$, $2$, $3$, $4$ and $1'$, $2'$, $3'$, ...
4
votes
1answer
154 views

Understanding the Definition of the Axiom Schema of Specification

Consider the Axiom Schema of Separation: If $P$ is a property (with paramter $p$), then for any $X$ and $p$ there exists a set $Y = \{u \in X : P(u,p)\}$ that contains all those $u \in X$ that ...
2
votes
2answers
57 views

Question Regarding the Replacement Schema

For each formula $\phi(x,y,p)$, we have the following axiom: $\forall x \forall y \forall z (\phi(x,y,p) \wedge \phi(x,z,p) \rightarrow y = z) \rightarrow \forall X \exists Y \forall y (y \in Y ...
5
votes
1answer
116 views

Where I can find the Pythagorean theorem deduced from Hilbert's axioms?

Hilbert took years to make a rigorous revision and formalization of Euclidean geometry in his Foundations of Geometry. As he intended to organize only the most basic aspects of the theory, he didn't ...
5
votes
3answers
285 views

Existence and uniqueness of God [closed]

Over lunch, my math professor teasingly gave this argument God by definition is perfect. Non-existence would be an imperfection, therefore God exists. Non-uniqueness would be an imperfection, ...
14
votes
2answers
155 views

How far is it true that statements dependent on Axiom of Choice are not constructive.

Axiom of Choice is often used in mathematics to construct various objects, such as basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, unmeasurable subset of $\mathbb{R}$, or a non-principal ...
3
votes
1answer
83 views

Success of Hilbert's Axioms

We know Euclid's axioms were found to be having many loopholes as in there were still many assumptions which weren't being stated in his system of axioms . Are Hilbert's axioms today completely ...
3
votes
1answer
72 views

ZF: Regularity axiom or axiom schema?

I have seen the axiom system ZF for set theory described including a single axiom of regularity (aka "foundation"), namely $$\forall x\neq\emptyset \, \exists y\in x \ y\cap x = \emptyset$$ and also ...
5
votes
3answers
158 views

Gödel's incompleteness wrt weakend versions of ZFC

Suppose for the sake of argument that we look at ZFC with the axiom of infinity removed. http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#First_incompleteness_theorem ...
5
votes
1answer
236 views

What set theory axioms do I need to believe in uncountable ordinals?

Math people: The title is the question. I am convinced that uncountable sets exist, thanks to Cantor's diagonal proof. It is not intuitively clear to me that uncountable ordinals or cardinals should ...
3
votes
2answers
147 views

Axiom of Regularity - Transitive set

I just managed to confuse myself completely while studying for Set Theory. We have the Axiom of regularity: $$\forall S (S\not= \emptyset \rightarrow (\exists x\in S)(S\cap x=\emptyset))$$ Now a set ...
1
vote
1answer
58 views

How to know when a system of axioms is 'complete'?

Here, I (basically) stated the group axioms as follows. $(xy)z=x(yz)$ $xe=x, ex=x$ $xx^{-1}=e$ In that post, answerers Martin and Ittay were critical of the above list for not including ...
6
votes
3answers
165 views

Question Regarding the Axiom of Extensionality

Jech's text on Set Theory states the following: If X and Y have the same elements, then X = Y : ∀u(u ∈ X ↔ u ∈ Y ) → X = Y. The converse, namely, if X = Y then u ∈ X ↔ u ∈ Y, is an axiom ...
9
votes
1answer
167 views

What is the smallest fragment of ZFC that has the same consistency strength as ZFC?

The question in the title is undoubtedly nonsensical, but I am not sure how to state this question properly. Perhaps some examples will help me explain it. Thanks to Godel and Cohen, we know that ...
1
vote
2answers
71 views

¿Can you help me with axiom of regularity? [duplicate]

I understand that if we define a set such that $A=\{A\}$ we have then a contradiction because of regularity axiom, thus, a set can not be a member of itself. My question is; how can it be so, if I ...
4
votes
2answers
103 views

ZF Extensionality axiom

To familiarize myself with axiomatic set theory, I am reading Kenneth Kunen's The Foundations of Mathematics that presents ZF set theory. I haven't gotten really far since I am stuck at the axiom of ...
3
votes
2answers
134 views

Axiom of infinity exceptions?

It is my understanding that the axiom of infinity, or axioms that depend on the axiom of infinity, are the only axioms that are not representable in finite First-Order Predicate Logic. Is this ...
3
votes
2answers
45 views

Schema of separation and set of all sets

The schema of separation states (this is probably simplified) states that if $P$ is a property and $X$ is a set, then there exists a set $Y = \{ x \in X : P(x)\}$. The notes I'm reading say that from ...
4
votes
3answers
466 views

A basic doubt on axiom of foundation of Zermelo-Fraenkel set theory

I have read in a book that the "axiom of foundation prevents anomalies such as a set being an element of itself". Now, axiom of foundation says that there exist an element in every set which is ...
6
votes
1answer
223 views

An Axiomatic Treatment of Mathematics from First Principles to the Major Subjects?

I'm looking for a book - more likely, books - that could take me from the axioms of mathematical logic up to the major subjects of mathematics, like analysis, algebra, geometry, etc. For example, a ...
1
vote
0answers
70 views

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 ...
2
votes
1answer
62 views

Are there examples of theorems proved via proper (i.e. non-conservative) extensions?

This is not a question about set theory specifically, but lets talk about ZFC just for concreteness Suppose we have a sentence $\phi$ in the language of ZFC, and a proof that either $(\mathrm{ZFC} ...
2
votes
1answer
86 views

how to show associativity of multiplication for not just 3 operands but for n operands

ie Id like to show a(bc)=(ab)c but for any n operands eg abcdefg=gfdcabe etc I can see this is very intuitive that this should be true for all n operands, but as a logical exercise I would like to ...
5
votes
3answers
234 views

Pythagorean theorem and its cause

I'm in high school, and one of my problems with geometry is the Pythagorean theorem. I'm very curious, and everything I learn, I ask "but why?". I've reached a point where I understand what the ...
2
votes
1answer
50 views

Infinite axiom schemas - what comes before?

The ZFC axioms constitute an infinite set. So too do the PA axioms. Presumably, then, we need a metatheory before we can even define these axiom systems. What is the usual metatheory? Do we even ...
2
votes
1answer
113 views

axiom of foundation of Zermelo–Fraenkel set theory

I have found two different statements on axiom of foundation of Zermelo–Fraenkel set theory in two different books as: 1) every nonempty set contains an element that is not an element of any other ...
3
votes
0answers
46 views

Have people studied weakened forms of Tarski's axiom?

My intuition about Grothendieck universes says the following. Axiom of empty set $\leftrightarrow$ There exists a Grothendieck universe. Axiom of infinity $\leftrightarrow$ There exist at least two ...
1
vote
0answers
58 views

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 - ...
5
votes
3answers
199 views

The axiom of infinity for Zermelo–Fraenkel set theory

The axiom of infinity for Zermelo–Fraenkel set theory is stated as follows in the wikipedia page: Let $S(w)$ abbreviate $ w \cup \{w\} $, where $ w $ is some set (We can see that $\{w\}$ is a ...
35
votes
8answers
3k views

Does mathematics require axioms?

I just read this whole article: http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf which is also discussed over here: Infinite sets don't exist!? However, the paragraph which I found most ...
2
votes
2answers
150 views

Formulation of the Separation Axiom in Set Theory

I have a question regarding this formulation of the Separation Axiom: I quote from the Book "Handbook of Mathematical Logic", the chapter from which this is taken could also be found here. Now we ...
5
votes
2answers
478 views

Is ZFC without Axiom of Infinity consistent?

The incompleteness theorem states that one cannot prove whether ZF or ZFC is consistent, but what about ZFC withouth Axiom of infinity? (Assuming the empty set exists) Furthermore, let $M$ be a ...
6
votes
2answers
119 views

Is there a way to axiomatize the category of sets and relations?

The system of axioms known as ETCS axiomatizes the category of sets and functions. Does anyone know of a way to axiomatize the category (and/or allegory) of sets and relations?
17
votes
5answers
601 views

Why hasn't GCH become a standard axiom of ZFC?

I've never seen a text that includes GCH in the ZFC axioms. I presume this means that GCH has not achieved widespread acceptance. This seems surprising to me, given that: The cardinal numbers ...
2
votes
2answers
119 views

Would it be possible to concoct a “harmful” axiom?

Suppose I run an automated theorem prover. It begins with the axioms of ZFC, and using a random number generator, it proves more theorems, and it runs for two days. At the end of the second day, it ...
1
vote
1answer
123 views

What makes Tarski Grothendieck set theory non-empty?

I'm fighting with Grothendieck set theory for some time now. This is the framework for the automated proof checking system of Mizar and hence there is a formalized version of the axioms here too, and ...
1
vote
1answer
86 views

Real numbers axiomatization without natural numbers

I think to remember that there is way to uniquely characterize the real numbers $\mathbb{R}$ via an axiom set. I wonder if this is possibly without introducing some notion of the natural numbers ...
14
votes
1answer
335 views

Is there an axiomatic approach to ordinal arithmetic?

I've always wondered, is there an axiomatic approach to the arithmetic of ordinal numbers? If so, I imagine it would be on par with set theory in terms of its proof-theoretic strength.
0
votes
1answer
59 views

Is there a systematic account of the number systems in the following 3x3 grid?

Consider the following sets of numbers, viewed as number systems with signature $(+,\times,\leq)$. Let $\mathbb{X} = \{1,2,3,\cdots\}$ denote the nonzero natural numbers. Let the completion of ...
4
votes
1answer
649 views

What are the differences between Hilbert's axioms and Euclid's axioms?

Euclid had his axioms. Why would we need Hilbert's modern axiomatization of Euclidean geometry? What are key differences between the two sets of axioms?
26
votes
7answers
1k views

Are there infinite sets of axioms?

I'm reading Behnke's Fundamentals of mathematics: If the number of axioms is finite, we can reduce the concept of a consequence to that of a tautology. I got curious on this: Are there infinite ...
6
votes
5answers
881 views

How to demystify the axioms of propositional logic?

How might I go about getting some intuition on the typical axiom schemes given for propositional logic? They seem rather mysterious at first glance. For example, these are taken from: ...