# What are the ramifications of the limitations of ZFC set theory?

In the Wikipedia article on Zermelo-Fraenkel set theory says that the theory sets out to formalize a notion of sets such that "all entities in the universe of discourse are such sets." It goes on to say that ZFC has some limitations:

• It can not include urelements, elements of sets which are not themselves sets.
• It can only include classes, groups of sets which are defined by a property which all their members share, indirectly.

What ramifications does this have on the application of ZFC set theory to philosophy?

• Its probably the limitations of ZFC that make it so useful foundationally... – Mozibur Ullah Feb 5 '14 at 22:21
• How do you apply ZFC to philosophy? I've seen a lot of philosophy written about ZFC; but I'm not sure what you mean by applying ZFC to philosophy. – user4894 Feb 5 '14 at 22:27
• @user4894: Badiou uses ZFC+forcing in his philosophy of politics – Mozibur Ullah Feb 6 '14 at 0:48
• You may have set theory with urelements: they do not "look differently" from "standard" ZFC. About "philosophical ramifications", a lot regarding phil of math (see Penelope Maddy's books), quite nothing (important) outside math. – Mauro ALLEGRANZA Feb 6 '14 at 10:01

ZFC is simple enough to axiomatise easily, and be accepted by the majority of mathematicians - some did not - and complex enough to ask good questions that have amenable answers. Once axiomatised and understood sufficiently well, one can look at more complex set theories using ZFC as a base.

One angle in more complex set theories is to think of the axiom of infinity. The usual one in ZFC is the simplest infinite axiom. There are a number of others - usually known as the large cardinal axioms - that arrange themselves in a linear hierarchy of proof-theoretic strength. That is you can use them to prove stronger statements, for example Gentzen used one to show that Peano Arithmetic, pace Godel, was consistent. This increase in the size of sets can be seen as including classes and more.

Another is loosening the axiom of extensionality, so that sets with exactly the same elements are not the same, but merely indiscernable (in Liebnizs terminology), or in tradtional mathematical terminology - isomorphic. This route is more fully developed in the topos-theoretic version of set theory or ETCS. Further, seeing that ETCS is a specialised topos, one can consider toposes themselves to be a generalised set theory, and it is noticeable that in the category of toposes that the terminal object is Set - which is telling for the foundational aspect of ZFC. But also, that the forcing construction introduced by Cohen takes a geometric form here - one can consider the Cohen topos which uses the topology of double negation - this is the construction that Godel/Gentzen used to embed classical logic into intuitionistic logic to model forcing.

Another angle is looking at Choice. ZFC actually is ZF+Choice. Choice has been an axiom that has had controversial implicatons since its introduction. Its at the root of non-measurable sets and of the Banach-Tarski paradox. Its denial leads to the idea of constructive mathematics. Where its not enough to simply assert the existence of some mathematical object - one must also construct it. Generally one also denies LEM - the law of the Excluded Middle. Here one should not operate with sets in classical logic but intuitionistic logic.

Interestingly enough, toposes, when seen as generalised set theories, have as their internal logic as intuitionistic logic. They also have a dimension, so higher toposes, and in fact infinite dimensional toposes have as their logic Martin-Lof dependent type theory - which ties it into ideas of computation. Further, a particular flavour of it, intensional Martin-Lof, can be interpreted as homotopy type theory, which then introduces geometry (aka homotopy) into type theory.

One notes in this that toposes tie in three strands of modern mathemtics in a native manner - logic, geometry & sets.

Finally, one notices that ZFC is a theory of sets together with first-order logic. One can exchange the logic for another. Intuitionistic logic has been mentioned above, but there is also paraconsistent logic (which is in some sense the dual of intuitionistic logic) which by denying the principle of explosion, allows inconsistent statements, somehow the inconsistencies are controlled so as not to affect each other. One then gets inconsistent set theory - and one gets quite interesting mathematical results - like one can assume the existence of a Universal set and the Russell Set - both of which leads to paradoxes in ZFC. As a further positive virtue, it settles the continuum hypothesis (in the negative).

All of the above is in the philosophy of mathematics.

In Philosophy at large, one notes that for example, the French Philosopher & Maoist, Badiou uses ZFC explicitly in his philosophy of Love, Politics & Art. He calls ZFC the ontology of truth. This is explained in his book, Being & Event, and in the Logic of Worlds he goes onto Topos theory to futher extend his line of thought. Quite how this works I'm non too sure of...

• Incredibly reductively/briefly: Badiou overloads the mathematical notions with philosophical-political ones (equality is probably the most obvious [example in his work and candidate for such an operation], but the 'void' or empty-set as 'foundational' is incredibly important as well given his Lacanian orientation) – Joseph Weissman Feb 6 '14 at 1:33
• I'm tempted to claim it's something like the inverse of Lautman's or Deleuze's approach to philosophy of mathematics (insofar as Badiou is concerned with elementary maths/logics/set-theory and almost as an afterthought considers 'higher' advanced maths; whereas real analysis/calculus/categories/sheaves etc are all over Deleuze and Lautman's work in many ways.) Cf Zalamea and there is also a new text on Deleuze and philosophy of math that looks interesting. – Joseph Weissman Feb 6 '14 at 1:36
• I'd understand Badiou more easily, if he said that he was using ZFC as a metaphor; but he explicitly states he doesn't; one may not take his disavowal seriously. But in the new brave world of generalised set theory - the foundational status of ZFC looks a little more suspect. In toposes it still retains it being being the terminal object - so foundational. But when one considers inconsistent set theory, which I am tempted to, I'm not sure how Badious avowal of the foundational status of ZFC add up. Of course historically, and sociologically it is ZFC that remains the foundational bar. – Mozibur Ullah Feb 6 '14 at 1:51
• @Weissman: the deleuzian text is interesting. Leibnizs idea of continuity as quoted in the text looks remarkably like the traditional definition used in undergraduate mathematics and usually attributed to Cauchy. – Mozibur Ullah Feb 6 '14 at 2:38