Tagged Questions
5
votes
2answers
72 views
Questions about the concept of Structure, Model and Formal Language
When we start to define mathematical logic (specifically, propositional, first order, and second order logic) we start defining the concept of a language. At the begining this is done in a purely ...
5
votes
3answers
112 views
The Language of the Set Theory (with ZF) and their ability to express all mathematics
Accordingly, the Language of Set Theory (in this case using $ZF$ axioms) is built up with the aim to express all mathematics. Now, I know that, for example, the construction of the numbers ($\mathbf{ ...
4
votes
1answer
69 views
Can we use proper classes in this way, to define a new infinity larger than |Ord|?
I believe there is a way to do this that makes sense, and I explain it below. I would like to know if I did some obvious mistake, or if the idea doesn't make sense for some reason I didn't figure it ...
20
votes
4answers
470 views
Is $\mathbb{N}$ impossible to pin down?
I don't know if this is appropriate for math.stackexchange, or whether philosophy.stackexchange would have been a better bet, but I'll post it here because the content is somewhat technical.
In ZFC, ...
1
vote
2answers
62 views
Language, set and sentential calculus
I'm trying to learn sentential calculus now and I'm very confused with the following thing.
In many books on logic I've found out that before working with sentential calculus itself we need the ...
5
votes
1answer
83 views
Basic question about encoding ZFC into PA
1) Are ZFC and PA arithmetic mutually interpretable if we extend PA to PA+A , where A is the set formulas of PA that result from the translation of the axioms of ZFC (or any large cardinal axioms ...
22
votes
7answers
1k 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 ...
10
votes
1answer
117 views
Any branch of math can be expressed within set theory, is the reverse true?
Set theory seems to have the property of being "universal", in the sense that any branch of math can be expressed on its language. Is there any other branch of math with this property?
I am asking ...
3
votes
2answers
90 views
Are axioms and rules of inference interchangeable?
There is an equivalence between cellular automata and formal systems, you can code one into the other and vice versa. But in the the cellular automata (CA) the rules of inference are fixed and are ...
3
votes
1answer
68 views
Hilbert's Program
I'm trying to understand David Hilbert's intention in creating a mathematics foundation.
Hilbert's program intention to confirm that we can find a finite set of axioms with ...
1
vote
1answer
53 views
Axioms of arithmetic and isomorphism
What does it mean to say that the axioms of arithmetic (Peano's or Dedekind's) are sufficient to characterize the abstract, mathematical structure of the natural numbers up to isomorphism?
2
votes
2answers
133 views
Beyond Goedel incompleteness and lack of soundness/completeness of higher-order logics
As I understand that there are at least two fundamental limits of the development of the mathematics:
1) Goedel incompeleteness theorems (or more clearly Church thesis) effectively says that there ...
5
votes
0answers
72 views
Functions and metafunctions
I didn't get any responses to this question the first time around, so I've tried rewriting parts of it. If there's anything glaringly wrong with the questions I'm asking, please leave a comment!
...
30
votes
9answers
2k views
Infinite sets don't exist!?
Has anyone read this article? Set theory
This accomplished mathematician gives his opinion on why he doesn't think infinite sets exist, and claims that axioms are nonsense. I don't disagree with his ...
1
vote
2answers
131 views
Hard proof concerning the periodicity of trigonometrical functions. Is that a challenge or just trivial
i want to know if exist or if you can develop or give me ideas of a proof to show that the least number for which sine is periodic is $2\pi$
$$\neg \{\exists n\in \mathbb{R} \wedge n < 2\pi: ...
