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Dec
30
awarded  Scholar
Dec
30
accepted The standard role of intuitive numbers in the foundations of mathematics
Dec
30
comment The standard role of intuitive numbers in the foundations of mathematics
After a number of readings, research and meditation I have read again your answer, and I found that it's the right one. I can finally accept it. No matters what is my philosophical position (I haven't any). What matters is there is some consensus in the starting point of mathematics. If we can accept that, in practice, all of us only can obtain the same list of entities as the methatmatical natural numbers, so it would be an acceptable starting point. On the other hand, it seems that there is not any formal system capable of capturing all the properties that intuitive natural numbers have.
Sep
24
awarded  Autobiographer
Jul
31
comment The standard role of intuitive numbers in the foundations of mathematics
+1: Thanks. At least you understand what about I am talking about.
Jul
18
comment The standard role of intuitive numbers in the foundations of mathematics
About what your answer: you put in doubt if the intuitive natural numbers have a rol in foundations. Yes they are, because in the foundations we have to build-up first order logic and set theory from "nothing". The Peano axioms or other constructions in set theory are becoming very "later". But to proof some metatheorems, natural numbers and recursion are used in several ways, the qualifier "finite" (in the sense of "counting up to some n") is used everywhere referring to the length of logical formulas, and so on. So, a primitive or intuitive use of natural numbers is everywhere.
Jul
18
comment The standard role of intuitive numbers in the foundations of mathematics
My question is not about whether the intuition is valid or not, but what are the rules accepted for intuitive natural numbers. I can accept primitive intuition as part of foundations, but the problem is that people is vague when they have to say what rules, properties or theorems are in a the "intuitive" domain and which are not. My question is precise, and I am looking for precisions that people in general don't take care enough to point out in the books.
Jul
15
comment The standard role of intuitive numbers in the foundations of mathematics
I took a look to the paper you added. It's interesting, but it's out of topic, since the paper of your advisor is working with internal models in set theory. The "consistency" results there, only can be "relative consistency" results, because if you really have at hand a model for set theory, this contradicts the Godel incompleteness Theorem. It is common to have results of relative consistency in model theory, but ZFC has nothing todo do in the proof of the two most important Godel's Theorems: Completeness Th. of 1st order logic, and Incompleteness Th. of Arithmetic.
Jul
15
awarded  Commentator
Jul
15
comment The standard role of intuitive numbers in the foundations of mathematics
The problem is that you are missunderstanding the use of "set" in model theory. A "set" in model theory is not a ZFC set, but a naive-set-theoretic set. This is, again, of intuitive nature. The user tomasz is being the right approach to the topic. Please read his comments and check if you properly understand what we are talking about.
Jul
15
comment The standard role of intuitive numbers in the foundations of mathematics
OK, but we have "foundations" magazines, with papers there. The referees have to judge with some criteria. If I use a naive-N property to proof something in metatheory, how the referee knows if this particular step is right or not?
Jul
15
comment The standard role of intuitive numbers in the foundations of mathematics
Agree with you in all, except by the last phrase. Mathematics is science, or part of the science. It needs rigor, because is the language of a lot of important scientifical brances. Anyway, going back to the intuition of natural numbers, you have put the things in the right way: the naive natural number theory is a primitive which mathematicians believe true. So, I can here restart my question: what is exactly the form and properties of this "naive" theory of natural numbers? When I use a property of N in a "foundations" paper, which criteria will use the referee to say can/cannot be used.
Jul
15
comment The standard role of intuitive numbers in the foundations of mathematics
I am not satisified. The problem here is that "finite" can be defined through the intuition of natural number, and natural number can be defined from the concept of "finite". The "controversial" point remains being essentially the same. If there are "controversial" issues in the "foundations", then mathematics is not a reliable science. A solid starting point is need. About the 2nd question, is part of foundations too, because there the properties of N are used to index variables in 1st order logic, or to give proofs by induction in the length of formulas.
Jul
15
comment The standard role of intuitive numbers in the foundations of mathematics
I think it is important to understand that we assume ZFC axioms in order to do logic. This is simply wrong. You are not well informed. Godel Completeness Theorem doesn't need any ZFC at all, and it is a Theorem about 1st order logic.
Jun
29
comment The standard role of intuitive numbers in the foundations of mathematics
Define "define".
Jun
29
comment The standard role of intuitive numbers in the foundations of mathematics
@William Hilbert: If your asking me for a definition of intuitive natural number, it means that you are missing the point. The upvotes you obtained in your comment are wrong, also.
Jun
29
revised The standard role of intuitive numbers in the foundations of mathematics
edited tags
Jun
24
comment The standard role of intuitive numbers in the foundations of mathematics
Hilbert and Godel did not accept the framework of intuitionism, but they accepted the existence of natural numbers in an intuitive manner. So, I think that this answer is not adequate. Besides, you say that "it is inevtiable that you start with accepting the framework of intuitionism/constructivism...". To just "accept" things is act of faith, it is not science. Why do I have to accept it? What is implied in that "acceptance"? The lack of 3rd exluded rule in intuionism it seems as having nonsense by the major part of mathematicians nowadays.
Jun
22
awarded  Promoter
Jun
20
awarded  Student