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Are there other approaches for the foundations of mathematics, other than logic and set theory?

And why does set theory begin talking about objects and groups of objects. Is it proven somewhere that that is the most fundamental concept? What is the main idea behind set theory? I do understand it, but I try to get the bigger picture, in what way does it try to set up the foundations of mathematics? It would be good to compare it to other approaches.

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    $\begingroup$ You may consider also Category Theory $\endgroup$ – Mauro ALLEGRANZA May 22 '14 at 11:12
  • $\begingroup$ Is like a complete alternative to set theory? I mean can I replace set theory completely with category theory as the foundation? $\endgroup$ – user3111311 May 22 '14 at 11:14
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    $\begingroup$ See the Intro to the SEP entry referenced above : "Category theory is an alternative to set theory as a foundation for mathematics." $\endgroup$ – Mauro ALLEGRANZA May 22 '14 at 11:19
  • $\begingroup$ Oh that is amazing. @MauroALLEGRANZA in very short, how do you compare them? Is category theory in some way a more sound foundation? $\endgroup$ – user3111311 May 22 '14 at 11:24
  • $\begingroup$ I simply do not know how to answer ... what does it mean "more sound" ? I imagine that you are in some sense "unsatisfied" with some "philosophical aspects" of set theory (from a mathematical point of view, set-t works quite well ...); thus you have to go in deep into those aspects and compare with cat-t point of view. In the SEP entry there is a section devoted to Philosophical Significance [of Cat Theory]: can be a useful starting point. $\endgroup$ – Mauro ALLEGRANZA May 22 '14 at 11:51
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On one hand, how much more basic and foundational can you get than objects and sets of objects? Of course, I say that as someone who comes squarely from the logic and set theory camp. On the other hand, you do ask a really good question, especially if you want to consider approaches to objects and sets other than the Zermelo-Frankel axioms. For example, you may want to look at Russell's type theory or Quine's New Foundations. Actually getting away from explicitly talking about objects and sets, though, the only thing I can think of is category theory, and in particular topoi. That may seem like a bit of a cheat, though -- the idea you start with in topoi is to consider the elementhood relation for sets as an "arrow" in the category sense, so you don't really get away from sets as much as just model them differently.

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  • $\begingroup$ Very useful, +1! $\endgroup$ – user3111311 May 22 '14 at 11:26
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    $\begingroup$ If you are to suggest type theories then SEAR and HoTT should be mentioned as well; and if you mention alternative set theories then ETCS should be mentioned. $\endgroup$ – Asaf Karagila May 22 '14 at 11:30
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    $\begingroup$ Yes, there are lots of alternatives to ZF, I just wanted to give a couple of the more historically obvious examples as a place to start. $\endgroup$ – user128390 May 22 '14 at 11:35
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I will try to answer just one of the questions:

in what way does [set theory] try to set up the foundations of mathematics?

It should help to see specific examples of how concepts from other areas of math can be formalized in set theory. Let's consider two examples.

(1) A group can be considered as an ordered pair of objects $(G, f)$ where $G$ is a set and $f$ is a function $G \times G \to G$ satisfying certain logical properties. The Cartesian product $G \times G$ is the set of ordered pairs of elements of $G$. A function $G \times G \to G$ is a subset of the Cartesian product $(G \times G) \times G$ satisfying certain properties. The notion of "ordered pair" can be formalized in terms of (unordered) sets using Kuratowski's definition.

(2) A real number can be defined as a set of rational numbers satisfying a certain logical property (that of being a Dedekind cut.) A rational number in turn can be defined as a set of ordered pairs of integers, namely an equivalence class under a certain equivalence relation. Similarly, integers can be defined as equivalence classes of ordered pairs of natural numbers. Natural numbers can be defined as certain kinds of von Neumann ordinals, which are sets.

In both cases (1) and (2) the reductions to sets are not very elegant, but they get the job done. I think it is remarkable that we are able to do such a thing at all. You could try to reduce set theory and group theory to analysis instead, or perhaps reduce analysis and set theory to group theory, but I don't think you would find as much success.

Of course I don't claim to have shown here that some other approach to the foundations of mathematics, such as category theory, would not work just as well.

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The main idea of set theory, as I see it: At the very least, a set theory presents (1) a fixed number of ways that you can infer the existence of a set given the supposed existence of another set(s) (e.g. axioms for subsets, power sets, Cartesian products), and (2) definitions of equality for sets and ordered n-tuples.

Using such axioms, along with the rules of logic, it is possible, in theory, to derive all of modern number theory and classical analysis starting with nothing more than the presumed existence of just one Dedekind-infinite set.

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