1
vote
2answers
69 views

How to deal with equivalences in proofs?

There is a part I need clarification on regarding the use of equivalence and its symmetry. From what I understand in regards to symmetry is that: $ (p \equiv q) \equiv (q \equiv p) $. Given p and q ...
5
votes
1answer
114 views

Does every mathematical principle have a proof?

My question actually narrows down to the meaning of mathematical principle. While I'm looking for some principles, they usually have their proofs, so I thought "principle" has the same meaning as ...
-2
votes
2answers
60 views

Can a set of theorem be derived from different sets of incompatible axioms? [closed]

Let's take T = {set of statements} that can be derived from S1 = {set of axiom}. Assuming that we keep the same derivation rules, are there any other S, with S1 & S always false, from which we can ...
1
vote
0answers
35 views

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 $V = \{\mathbb N, +, =, X ,(,)\}$. ...
2
votes
2answers
64 views

Axiom of unrestricted comprehension

I'm doing some research on naive set theory and was a little confused over the statement of the axiom of unrestricted comprehension, $\exists$B$\forall$x(x$\in$B$\iff$$\phi$(x)). I am curious as to ...
4
votes
3answers
56 views

Examples of when $0\cdot x\neq 0$ or $x+0\neq x$.

I'm trying to prove that the arithmetic axioms are independent by constructing a model in which all bar one of the axioms are satisfied, for each of the axioms below. A first order theory with ...
1
vote
2answers
69 views

How an axiomatic system is made?

An axiom is a sentence that is taken to be true without a proof. A set of (well organised) axioms is called an axiomatic system. As consequence of these axioms we get a lot of results that we call ...
5
votes
1answer
274 views

How can I prove that there is no set containing itself without using axiom of foundation?

I've already found some similar questions in here (and other sites), but in most of the case, the use of axiom of foundation is required to complete the proof. Is there any way to prove $\not\exists ...
2
votes
2answers
59 views

Is it possible to formalize areas such as image processing and computer vision?

Is it possible to formalize areas such as image processing? By formalize I mean setup axioms, then derive theorems, and reason about image processing concepts and methods formally. I would say now ...
1
vote
1answer
74 views

How to extract the sets “produced” by a axiomatic set theory, into some newly introduced collection?

Say I know well how to reason in a set theory, which for the sake of this question I'll call $\bf{ST}$, say one of those, it can by $\bf ZFC$. It's principally written down via first order logic and ...
3
votes
1answer
62 views

How do we decide which axioms are necessary?

I am studying the axioms for a complete ordered field. I have looked at different sources, some of which differ slightly in their listings. Given some construction of the reals (e.g. Dedekind cuts or ...
1
vote
1answer
52 views

Closed under an operation?

So the question is write down a definition for it means to be closed under an operation. I said that a set is closed under an operation if that operation returns an element in the set when evaluated ...
5
votes
3answers
307 views

Derive by modus ponens $[A\rightarrow(B\rightarrow C)]\rightarrow[(A\rightarrow B)\rightarrow(A\rightarrow C)]$

How could I derive by modus ponens the formula $$[A\rightarrow(B\rightarrow C)]\rightarrow[(A\rightarrow B)\rightarrow(A\rightarrow C)]$$ from, and just from, the following axiom schemata? $(A\lor ...
4
votes
1answer
123 views

Can one define $\langle x,y\rangle$ in $P(C)$?

I study at course Foundations of Mathematics the below definitions and lemma: $\langle x,y\rangle:=\{\{x\},\{x,y\}\}$ (from Kuratowski 1921) $\langle x,y\rangle:=\{\{\{x\},\varnothing\}\{\{y\}\}\}$ ...
4
votes
2answers
177 views

What is the relation between axiomatic set theory and logical quantifiers?

On the one hand, the logical predicates $\forall$ and $\exists$ are defined using the concept of a Domain of Discourse, which itself is defined as a set (at least according to wikipedia). On the ...
7
votes
3answers
235 views

Why is zero important?

I am not sure whether this question is more appropriate here or in theoretical computer science. I leave it to the wisdom of moderators. On the computer science site I came across the following ...
5
votes
3answers
157 views

Axiom schema of specification (formula arguments)

Some sources define the formula like this: $$ \forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \wedge \varphi(x, w_1, \ldots, w_n , A) ] ) $$ Why ...
1
vote
2answers
123 views

Automatic theorem prover for proving simple theorems?

Is there a simple software that I could use to practice proving theorems in my course of mathematical logic? Basically what I need is ability to 1) define what axioms and laws I am allowed to use in ...
3
votes
2answers
171 views

axioms of equality

Every text I've come across uses the Axiom of Extensionality to describe the fact that sets are equal iff they contain each other's elements. $$(\forall x)(\forall y)\bigl((\forall a)(a\in x \iff a\in ...
1
vote
1answer
156 views

Explain mathematical practice and axiomatization to non-mathematicians

I am asked to give a talk about (a) mathematical practice, (b) axiomatization, (c) Gödel's theorems and (d) possible antimechanist arguments based on the incompleteness theorems (as mentioned in P ...
2
votes
2answers
109 views

How can we define infinitary proofs?

In the first order logic the usual notion of a formal proof for a sentence $\sigma$ from a theory $T$ is a "finite" sequence ($<\omega$ - sequeance) of sentences which each one of them is a valid ...
3
votes
1answer
136 views

Axioms based on $\leftrightarrow, \lor, \bot$ for propositional intuitionistic logic?

Propositional intuitionistic logic can be axiomatized based on $\;\to, \land, \lor, \bot\;$, with modus ponens $$ \text{from }\; \phi \;\text{ and }\; \phi \to \psi \;\text{ infer }\; \psi $$ as the ...
15
votes
2answers
232 views

How much math do we need to prove all simple numeric identities?

Consider real numeric expressions build only from integers, operators $+,-,\times,/$ and taking a positive expression to a power (no variables involved), e.g. $$\frac{2}{7},\ 2^{1/2},\ ...
4
votes
1answer
144 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 ...
2
votes
1answer
54 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
votes
1answer
61 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
votes
2answers
114 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?
0
votes
1answer
123 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
votes
5answers
693 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 ...
0
votes
1answer
83 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 ...
10
votes
6answers
279 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 ...
0
votes
4answers
88 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
vote
1answer
90 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 ...
11
votes
2answers
526 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?
2
votes
1answer
225 views

Axioms and deduction rules of Propositional Calculus

I'm looking for a list of axioms and fundamental deduction rules of Propositional Calculus.
4
votes
1answer
150 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 ...
5
votes
3answers
281 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, ...
5
votes
3answers
156 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 ...
6
votes
1answer
219 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 ...
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 ...
5
votes
1answer
430 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 ...
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
831 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: ...
6
votes
2answers
352 views

How does (ZFC-Infinity+“There is no infinite set”) compare with PA?

How does (ZFC-Infinity+"There is no infinite set") compare with (first order) PA? Intuitively, neither theory should be more powerful than the other.
2
votes
2answers
115 views

What percentage of formulas is unprovable in a given axiomatic system?

I am trying to use language I am not familiar with, so bear with me. If I make no sense, I try to be clearer. Assume we are given a formal language. Assume $S$ is the set of every well-formed formula ...
3
votes
1answer
192 views

System with infinite number of axioms

Assume we have a set of axioms $A_0$. There exists a statement that can be formulated with these axioms that cannot be proven to be true with this system. Assume we give such a statement axiomatic ...
1
vote
2answers
182 views

How can we know arithmetical axioms are consistent?

If we assume both distributivity and the opposite of the law of signs (ie, that $-1\times-1 = -1$) for the relative integers, then we can derive that two different numbers are actually equal. ...
2
votes
5answers
218 views

Is there an axiom that prevents other axioms from contradicting each other?

i.e. Does an axiom already exist, which prevents the addition of those new axioms which can contradict already existing axioms? Also, who decides that something is an axiom?
3
votes
2answers
146 views

How bad is this analogy for logical independence?

It is an amazing and well-known fact that the Continuum Hypothesis is logically independent of Zermelo-Frankel set theory with the Axiom of Choice (ZFC), assuming it is consistent. In a similar vein, ...
3
votes
3answers
374 views

Why is axiomatic system needed in propositional logic?

I am trying to learn propositional logic. I have read that axiomatic system is defined since there are some problems which can not be solved using truth tables. I have found such a problem in ...