Every foundational system for mathematics I have ever read about has been a set theory, from ETCS to ZFC to NF. Are there any proposals for a foundational system which is not, in any sense, a set theory? Is there any alternative foundation which is not a set-theory?
[Math] Set theory and alternative foundations
set-theory
Related Solutions
One can do this using less technology, too...
Let $\Sigma$ be an alphabet, $N$ a set of non-terminals, and $\Sigma^\*$ and $(\Sigma\cup N)^\*$ the full languages on $\Sigma$ and $\Sigma\cup N$, respectively. A context-free grammar is a finite subset $G\subset N\times(\Sigma\cup N)^\*$. Given one such grammar $G$ there is a relation $\mathord\rightarrow_G\subseteq(\Sigma\cup N)^\*\times(\Sigma\cup N)^\*$ which is the least transitive reflexive relation which contains $G$ (notice that $N\times(\Sigma\cup N)^\*\subseteq (\Sigma\cup N)^\*\times(\Sigma\cup N)^\*$, so this makes sense) and such that $$a\rightarrow_Gb \wedge a'\rightarrow_Gb'\implies ab\rightarrow_Ga'b'.$$ The language generated by $G$ from a non-terminal $n\in N$ is just $L(G, n)=\{w\in \Sigma^\*:n\rightarrow_Gw\}$. This is, in fact, the standard way to do this...
Your worries arise from asymmetry between how you view ordinary mathematics and how you view logic and model theory.
If it is the business of logic and model theory to provide foundations for the rest of mathematics then, of course, logicians and model theorists will not be allowed to use mathematical methods until they have secured them. But how might they accomplish this? The more we think about it, the more it becomes obvious that "securing the foundations of mathematics", whatever that means, is a task for philosophers at best and a form of mysticism at worst.
It is far more fruitful to think of logic and model theory as just another branch of mathematics, namely the one that studies mathematical methods and mathematical activity with mathematical tools. They follow the usual pattern of "mathematizing" their object of interest:
- observe what happens in the real world (look at what mathematicians do)
- simplify and idealize the observed situation until it becomes manageable by mathematical tools (simplify natural language to formal logic, pretend that mathematicians only formulate and prove theorems and do nothing else, pretend that all proofs are always written out in full detail, etc.)
- apply standard mathematical techniques
As we all know well, the 20th century logicians were very successful. They gave us important knowledge about the nature of mathematical activity and its limitations. One of results was the realization that almost all mathematics can be done with first-order logic and set theory. The set-theoretic language was adopted as a universal means of communication among mathematicians.
The success of set theory has lead many to believe that it provides an unshakeable foundation for mathematics. It does not, at least not the mystical kind that some would like to have. It provides a unifying language and framework for mathematicians, which in itself is a small miracle. Always remember that practically all classical mathematics was invented before modern logic and set theory. How could it exist without a foundation so long? Was the mathematics of Euclid, Newton and Fourier really vacouous until set theory came along and "gave it a foundation"?
I hope this explains what model theorists do. They apply standard mathematical methodology to study mathematical theories and their meaning. They have discovered, for example, that however one axiomatizes a given body of mathematics in first-order logic (for example, the natural numbers), the resulting theory will have unintended and surprising interpretations (non-standard models of Peano arithmetic), and I am skimming over a few technical details here. There is absolutely nothing strange about applying model theory to the axioms known as ZFC.
Or to put it another way: if you ask "why are model theorists justified in using sets?" then I ask back "why are number theorists justified in using numbers?"
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Best Answer
Bill Lawvere has suggested axiomatizing the category of categories as a foundation of mathematics, and there is no sense in which this could be thought of as a set theory. Colin McLarty is one person who has done some work on achieving such an axiomatization.