1

In textbooks on logic the authors usually use the notions of set and map immediately, long before the set theory is constructed. That is strange for me, and I want to ask if anybody can advise me a textbook on logic with a "linear structure", without these "circles in definitions"?

EDIT. This question is a reference request, so I hope, if I cross post it to MathOverflow, this will not cause a duplication of efforts for people who plan to post answers. Here is this question at MO.

Martin Sleziak
  • 53,687
Sergei Akbarov
  • 2,445
  • Carl Mummert's answer here is quite relevant. – hmakholm left over Monica Sep 29 '16 at 19:32
  • It is not possible in an "absolute" sense: imagine writing a grammar to describe "how language works". Do you think that you can do it without using language in writing the book ? – Mauro ALLEGRANZA Sep 29 '16 at 19:35
  • 1
    But you can see Kenneth Kunen, The Foundations of Mathematics (2009). – Mauro ALLEGRANZA Sep 29 '16 at 19:40
  • @HenningMakholm, yes, I am not against Carl Mummert's idea of "describing a preliminary set theory" for using it in propositional calculus and in predicate calculus before the "true set theory" will be constructed. Did anybody do this in a textbook? – Sergei Akbarov Sep 29 '16 at 20:10
  • @MauroALLEGRANZA, as I can see, Kenneth Kunen describes set theory without mentioning axioms of first order logics that he uses. In my opinion that is not good. Do there exist texts where set theory is constructed as a first order theory with specifying the axioms of the corresponding first-order logic? Actually, I don't see the problem: one can formulate the axioms of predicate calculus + axioms of set theory, then consider the corollaries, and only after that discuss what they call "semantics of logic"? What is wrong with this? – Sergei Akbarov Sep 29 '16 at 20:16
  • @SergeiAkbarov there is a book called "lógica y teoría de conjuntos" written by Carlos Ivorra Castillo, which you can download here but it is in spanish. That book is divided in three parts, the first one is all about first order logic, the second one is about the logic behind set theory and finally the third one is about set theory. Sadly I don't think there is an english version, but if you understand spanish you might find it useful – la flaca Sep 29 '16 at 21:54
  • @Eliana, unfortunately, I don't speak Spanish. If there is something in English or French, I would appreciate very much! – Sergei Akbarov Sep 29 '16 at 22:12
  • Does Russell and Whitehead's Principia Mathematica count? It sounds like you are looking for a completely formal account of mathematics. On a more practical note, if your only problem with Kunen's book is that he doesn't define the axioms of first-order logic, you can find another account of that elsewhere. There are proof theory texts which present FOL in a completely syntactic sense. – Jason Rute Sep 30 '16 at 15:29
  • @JasonRute, I thought, Russell and Whitehead are not used now, that isn't true? Which texts on FOL do you mean? – Sergei Akbarov Sep 30 '16 at 15:53
  • @SergeiAkbarov, you never specified your intended use. Yes, Russell and Whitehead's book is never used in practice. However, it was a very clear attempt to give this linear structure you have in mind. For pedagogical purposes, it is difficult to present foundations from the ground up, and almost no (if any!) textbook does it. There are formal treatments of mathematics on computers that are completely formal, but this is not suitable to "read" as a textbook. – Jason Rute Sep 30 '16 at 16:04
  • 1
    @SergeiAkbarov, if you are looking for a treatment of FOL with syntax and not semantics, then you could read the first section of these notes (http://math.ucsd.edu/~sbuss/ResearchWeb/handbookI/ChapterI.pdf). The semantics stuff comes later and can be skipped. (I am sure there are books too, but I don't have a good source off hand. The one I know of is still being written.) – Jason Rute Sep 30 '16 at 16:10
  • 1
    For people interested in constructive foundations, here are some relevant links: http://math.stackexchange.com/a/1808558, http://math.stackexchange.com/a/1895288. I should note that the question of whether LEM (law of excluded middle) holds is not clear to me beyond the arithmetical hierarchy, even granting the complete collection of the natural numbers, because the original intuition or justification for LEM only applies to assertions about reality (see http://math.stackexchange.com/a/1888389). – user21820 Oct 01 '16 at 12:06
  • @JasonRute, thank you, I am reading this. – Sergei Akbarov Oct 02 '16 at 12:59
  • @user21820, thank you too. – Sergei Akbarov Oct 02 '16 at 13:02

0 Answers0