0
votes
1answer
56 views

particular property and completeness?

I was puzzeling with the almost standard definition of completeness: In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula ...
5
votes
2answers
286 views

Who stole the axioms in Natural Deduction?

The study of Gentzen's sequent calculus give me the opportunity to propose some reflections about the concept of logical truth. I'll refer to the english edition of Gentzen's works : The collected ...
5
votes
4answers
369 views

What is the “correct” reading of $\bot$?

I have some doubts about the "natural" interpretation of $\bot$ in Natural Deduction and sequent calculus. In Prawitz (1965) $\bot$ (falsehood or absurdity) is called a sentential constant [page 14] ...
2
votes
1answer
171 views

True and provably true sentences in a model. Are they the same thing?

In logic, it is said that each sentence in a (consistent) theory is either true or false in a given model. Checking the truth of a sentence in a finite model amounts essentially to finite enumeration ...
4
votes
5answers
467 views

Books on logic, proof theory and set theory?

I graduated in Computer Science at University of Bologna in Italy some years ago. For various reasons now I am discovering a back interest in mathematic logic higher than I was a student. I have only ...
8
votes
2answers
875 views

Definition of “non-constructive proof”

I was wondering if it is possible to define exactly what a non-constructive (nc) proof is. I have often seen the concept associated with the use of principles such as the axiom of choice or the law of ...
6
votes
2answers
269 views

Ideas about Proofs

If there are two different proofs for one theorem, at some level are the two proofs the same, or can they be fundamentally different? In other words, if you have two proofs of a theorem, can one show ...
11
votes
7answers
1k views

Why do statements which appear elementary have complicated proofs?

The motivation for this question is : http://math.stackexchange.com/questions/4066/rationals-of-the-form-fracpq-where-p-q-are-primes-in-a-b and some other problems in Mathematics which looks as if ...
27
votes
6answers
2k views

If all sets were finite, how could the real numbers be defined?

An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who ...