1
vote
1answer
66 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 ...
2
votes
1answer
49 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
43 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 ...
4
votes
4answers
257 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
120 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
158 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
222 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
136 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
88 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
107 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
148 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
93 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
129 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
213 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
127 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
51 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
59 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
111 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
113 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
531 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
258 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
83 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
88 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
471 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
214 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
143 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
273 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
153 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
182 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 ...
32
votes
7answers
2k 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 ...
4
votes
1answer
376 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 ...
25
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
668 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: ...
5
votes
2answers
326 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
108 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
168 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
178 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
214 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?
-1
votes
1answer
98 views

Arbitrary definition [closed]

I find it most perplexing for how the definition of say, a set, defines itself. How does it become tangible? Where does it come from? Because, arbitrary definition seems more so like just saying "it ...
3
votes
2answers
141 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
330 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 ...
0
votes
1answer
106 views

A substitution inference in predicate calculus

I'm not sure what I'm missing here, but I have this lemma I am trying to prove, and it is giving me a lot of trouble. I'm technically working in ZF set theory, but this part doesn't need much more ...
21
votes
6answers
822 views

Axiomatic characterization of the rational numbers

We have the well-known Peano axioms for the natural numbers and the real numbers can be characterized by demanding them to be a Dedekind-complete, totally ordered field (or some variation of this). ...
3
votes
2answers
644 views

What is the difference between an axiom and a postulate?

I here about axioms is set theory and postulates in geometry, but they seem like the same thing. Do the mean the same thing but then are used in different instances or what? Is one word more ...
2
votes
2answers
750 views

Is Gödel's theorem invalid? [closed]

Right now I've skim through Gödel's theorem is invalid by Diego Saá on arXiv (freely available). As it seems very plausible, I ask for any references and scrutinizations of the paper.
4
votes
3answers
318 views

Please help. Great confusion about the "two levels of discourse'' mathematical logic.

I regard mathematics as being build up in the following way: We have some collections of symbol and rules (which are and have to be described in a natural language) to manipulate these symboles. If we ...
8
votes
2answers
246 views

Axiomatic system and Hilbert's 2nd problem

Hilbert's second problem asks if the axioms of arithmetic are consistent. Has this problem been resolved? Shouldn't an axiomatic system ideally be consistent and complete(given that we have the ...
-1
votes
2answers
458 views

Axioms of set theory and logic

Zermelo–Fraenkel set theory is the most common foundation of mathematics with eight axioms and axiom of choice (ZFC): http://plato.stanford.edu/entries/set-theory/ZF.html But one can see that the ...
40
votes
9answers
12k views

How is a system of axioms different from a system of beliefs?

Other ways to put it: Is there any faith required in the adoption of a system of axioms? How is a given system of axioms accepted or rejected if not based on blind faith?