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

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1answer
158 views

Describe a sound and complete proof system

I have a homework assignment that I am a little stumped on, the questions is: Describe a sound and complete proof system (axioms and proof rules) for proposition logic. Explain in detail why you ...
3
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2answers
110 views

Does the language of group theory need a constant?

I would like to verify the understanding of my claim. In Model Theory: An Introduction by David Marker Example 1.2.5 defines the language of the theory of groups as follows to be $\mathcal{L}=\{., ...
13
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1answer
474 views

Can we prove the existence of $A\cup B$ without the union axiom?

If $A$ and $B$ are sets, then $A\cup B$ is defined as follows: $$A\cup B:=\{x: x\in A \, \, \,\text{ or } \, \, \, x\in B\}.$$ In ZFC, $A\cup B$ exists because $\bigcup\{A,B\}$ exists by union axiom, ...
3
votes
1answer
109 views

Deriving the Axiom of Infinity

I'm currently reading about recursion, working on various exercises since I haven't formally studied the material before. The main book that I'm following is Kunen's Set Theory book There's an ...
0
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0answers
250 views

How are multiplication and addition defined?

Many people argue that multiplication is not necessarily repeated addition, indeed in the field of order 4 given on Wikipedia, adding one to itself A times will never yield A*1. See: ...
2
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2answers
193 views

Can multiplication be defined without addition?

I'm struggling to understand how to define multiplication and addition, now that I've been told that multiplication is not just repeated addition. It seems that the axioms for the two are ...
5
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1answer
159 views

What if we change axiom of arithmetic?

I want to experiment with some ideas about changing the axioms of arithemetic. (You know, like changing axioms of Euclidian geometry, we get non-Euclidian geometry.) What happens when we change the ...
3
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2answers
120 views

Help with a proof by induction.

I'm reading On Mathematical Induction by Leon Henkin in JSOTR. And well, in the first part of the article the author given us the necessary properties to be a Peano Model. A model $\langle ...
2
votes
1answer
55 views

Are forcing techniques possible to automate for mechanized reasoning?

Looking at Cohen's success at proving independence of the Axiom of Choice and the Continuum Hypothesis, I was wondering if it was possible to mechanize forcing techniques for the purpose of proving ...
5
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1answer
63 views

Weak classical Deontic Logics

I am writing a paper at the moment and an area of Deontic Logic has cropped up in it. I know very little about the area and I was wondering if people could give me opinions on the axiomatic system ...
3
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2answers
115 views

What is meant by one axiom being “weaker” than another?

If we have two axioms $A$ and $B$, what exactly is meant by axiom $A$ being weaker than axiom $B$? This question is a follow-up to A weaker Axiom of Infinity?
5
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1answer
176 views

A weaker Axiom of Infinity?

As I understand, the Axiom of Infinity in ZFC theory gives us an infinite set from which the natural numbers can be extracted. (See "Axiom of Infinity" at ...
0
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2answers
132 views

what are the Rosser Turquette axioms of Lukasiewicz 3 valued propositional logic?

I am trying to get my head around the Rosser-Turquette axiomatisation of Lukasiewicz n-valued logics, but cannot really follow it. Maybe if somebody can give me the axioms for 3 and 4 valued logic ...
2
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5answers
872 views

Are (some) axioms “unprovable truths” of Godel's Incompleteness Theorem?

Like any math newbie, Godel's Incompleteness Theorems are easy to understand in general layman's terms, but difficult to understand beyond the typical "liar's paradox" and "barber's paradox" type ...
5
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5answers
145 views

Help understanding $x=y\Rightarrow(x=z\Rightarrow y=z)$

I was reading a proof that opened with the integer axiom of $x=y\Rightarrow(x=z\Rightarrow y=z)$ What would be an accurate statement in English to express this? The "implies" within the first ...
2
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2answers
84 views

What is the absolute minimum that must be accepted/defined in order to prove 1+1=2?

As I know the "1 + 1 = 2" question has been asked before, my question (for now anyway) is more specific: Assuming as little as possible and accepting as few definitions as possible, what are the very ...
2
votes
1answer
98 views

weaker version of some axioms

I'm complementing my knowledge of set theory using other sources in addition to the book that normally I use. Working with the Jech's book (great book, by the way), in one exercise of the first ...
0
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1answer
84 views

Proving statements for individual and/or all sets of axioms

If something is proved for one set of axioms then you can use it for that set of axioms but wouldn't you have to prove it for another set of axioms before you could use if for the second set of ...
6
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4answers
350 views

Why the axioms for a topological space are those axioms?

This question might have even been asked here before, I don't really know, so sorry if it's duplicate. I've started to study topological spaces and I've found the axioms for such spaces kind of hard ...
10
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6answers
302 views

Axiomatic Foundations

I am trying to deduce how mathematicians decide on what axioms to use and why not other axioms, I mean surely there is an infinite amount of available axioms. What I am trying to get at is surely if ...
2
votes
4answers
244 views

Definition of $0$?

In real number axioms, it is defined that there is $0$ such that $x+0=x$ for all $x.$ I was wondering an example is there any other algebraic structure than real numbers which satisfy the real number ...
0
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4answers
92 views

Replacing Axiom of Extensionality with a logical formalism

Is it possible to replace the Axiom of Extensionality with a formalism from logic, namely the following one: $\forall a \forall b (a=b\Leftrightarrow \forall P (P (a)\Leftrightarrow P (b)))$ ($ P $ is ...
1
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1answer
92 views

Apparent equivalent notations for the axiom of infinity

I'm just begining to build the systems of numbers based on the axioms of set theory ($\mathsf{ZF}$). Accordingly the axiom of infinity is no more than assuming the existence of $\mathbb{N}$ (of course ...
3
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3answers
140 views

If we abandon the axiom of regularity, can the cumulative hierarchy just become a definition?

In ZFC, the axiom of regularity is used to prove that every set is an element of some stage of the cumulative hierarchy. The index of the least such stage is, by definition, the rank of that set. Now ...
2
votes
1answer
64 views

Can I express (only) the syntactical formulation of a set theory within another.

Motivated by this question on the SE philosophy board "Is there a one true set theory", I want to ask for a clarification: I've seen e.g. ZF being expressed in Grothendieck set theory. If I say ...
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3answers
110 views

Is there a consistent axiomatic system where $1+1 = 2$, $2 \not = 1+1$?

Is it possible to create a consistent axiomatic system where $1+1 = 2$, $2 \not = 1+1$?
3
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2answers
315 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
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0answers
167 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
566 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
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1answer
144 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
318 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
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1answer
237 views

Axioms and deduction rules of Propositional Calculus

I'm looking for a list of axioms and fundamental deduction rules of Propositional Calculus.
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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
246 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
136 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
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1answer
161 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
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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
124 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
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3answers
298 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
169 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
84 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
75 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
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3answers
161 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
253 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
170 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 ...
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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 ...
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3answers
170 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
175 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 ...
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vote
2answers
80 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
107 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 ...