The Language of the Set Theory (with ZF) and their ability to express all mathematics

Question

Accordingly, the Language of Set Theory (in this case using $ZF$ axioms) is built up with the aim to express all mathematics. Now, I know that, for example, the construction of the numbers ($\mathbf{ \mathbb{N},\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}}$) is got right from the axioms and definitions from the Language of Set Theory, so everything is inside the same theory. But, when we are dealing with especial structures like for example Euclidian Geometry, we need to state new axioms, and, apparently, it doesn't make any sense to add this axioms to our original set theory because it might be possible to find a different model, maybe completly artificial, in which some axiom of Euclidian geometry migth not hold or result in a contradiction. But Euclidian Geometry and all the different theories that one might think of are part of mathematics. So my point is that it is impossible to develop all of mathematics just from the axioms of set theory. Does it make sense? Please tell me if I'm wrong and if I'm missing something. Also I have these questions:

In building these new theories (like euclidean geometrie, etc), each theory uses the language of set theory. But also we add new axioms. So, are we talking of a different language? Are we talking of a different theory? Is the language of set theory a metalanguage in this new language? I'm totally confused.

Answer 1

The common interpretation of the claim that you can do maths in $ZF$ is that given a model of $ZF$ you can produce in it models of all relevant objects of study in mathematics. So, as you mention, given a model of $ZF$, you can construct in it a set which models the natural numbers, a set which models the real numbers, and so on. That does not mean that you add any new symbols for addition or multiplication, but rather that you show that there is set in the universe of sets of the $ZF$ model together with identifiable extra structure that turns it (while it's still just a set) and all that extra structure (still all just sets, since that is all you have in $ZF$) into a model of, say, the natural numbers satisfying Peano + Induction. Similarly for other number systems.

In the same way you can construct a model of Euclidean geometry as well as models of non-Euclidean geometry all within the confines of a $ZF$ model. It is in this sense that "mathematics can be done within $ZF$". Of course, it is possible that some aspects of mathematics can't be captured by $ZF$. For instance, category theory requires a tower of $ZF$ models with suitable properties (see Grothendieck universes).

You are in a sense correct that when modeling, say, the real numbers inside $ZF$, the language of set theory can be thought of as becoming a meta-language and the language of the reals becomes the actual language. But this is not how it's thought of in practice. In practice, when you deal with the reals for the purposes of real analysis you couldn't care less that they can be modelled inside $ZF$. Instead, you just work with them axiomatically, assuming a background language and model of (naive) set theory. The difficulties arise when one realizes that there are many different models of $ZF$ (unless of courser there aren't any models at all) and these may affect what happens in the reals. The study of the effects of certain axioms of set theory on the resulting models of the reals inside is part of axiomatic set theory and can get quite complicated. You can also have a look at topos theory where similar things happen. In topos theory the idea is to axiomatize some basic properties of the category of sets in such a way that such a construct admits an internal logic. Many toposes allow for internal constructions of natural numbers, real numbers etc. And funny things may happen. For instance, there is a topos in which the internal notion of real numbers is such that all functions from the reals to the reals are continuous.

Answer 2

First of all, let me point that the axioms of $\sf ZF$ were given to give a reasonable axiomatization for the concept of "set". That's all. And don't feel bad that you're confused, this is a very confusing point.

To address you actual question, in mathematics we have a theory, which is what we work with - e.g. Euclidean geometry - and we have a meta-theory which is the mathematical theory in which we work. This meta-theory allows us to perform certain things (for example, formalize the concept of a proof or a model).

In set theory we can describe what does it mean for something to be a language, that is a collection of symbols with certain rules. It's really just a set which the correct cardinality. And we can write the rules for creating formulas and sentences in that language, so we can write down the axioms -- for example the axioms of Euclidean geometry, or even the axioms of $\sf ZF$ itself.

Next we can describe what does it mean to be an interpretation for the language. It's a pair, a non-empty set (the universe of the structure) and the function which interprets every symbol in the language into a relevant element/subset/relation on that universe. And with these we can ask whether certain sentences are true in this structure. And we can ask whether or not there is a structure to the language of geometry in which the axioms of geometry.

But all this happens inside a universe of set theory. We don't add to the meta-theory, we work inside the meta-theory. We use the meta-theory like an ambulance. It takes us to the hospital, but we don't perform brain surgery inside it.

So really what we do is define some sets which represent our language, the axioms, the model, and we prove things on these sets which in turn end up to be proofs in and about Euclidean geometry. Or group theory. Or lattice-ordered monoids. It doesn't matter.

But the huge point, which I can't stress enough, is that $\sf ZF$ is the foundational theory. It's like a planet. When want to build a structure we don't add planets to the solar system, we just build a structure inside the planet Earth, using material that Earth has provided us with.

The same happens with $\sf ZF$ or $\sf ZFC$. The theory provides us a planet, and materials to do mathematics and to construct mathematical models and universes for different theories. All using sets and $\in$.

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Other links of interest:

1. When we say, "ZFC can found most of mathematics," what do we really mean?
2. How to write $\pi$ as a set in ZF? (In particular the comments to the answers are relevant to your question)

Answer 3

The way one can develop both Euclidean and non-Euclidean geometry in set theory is by considering the objects of geometry, such as points and lines, as kinds of sets. For example, it is natural to consider a line as a set of points. It is not so natural, but still possible, to consider a point as a set. It doesn't matter much how we do this, only that it is possible via various formalisms. For example, a point $(x,y)$ in $\mathbb{R}^2$ can be considered as the pair $\{\{x\},\{x,y\}\}$ according to Kuratowski's definition of the ordered pair in set theory, and then the real numbers $x$ and $y$ can be considered as sets of rationals according to the definition of $\mathbb{R}$ in terms of Dedekind cuts, and so on.

With a bit more work, one can describe models of non-Euclidean geometry such as the pseudosphere in terms of sets. One way to do this is to first embed these models into a higher-dimensional ambient Euclidean space. A more abstract way is to use the notion of a manifold, which can still be described purely in terms of sets, but this involves so many steps now that most people would not want to do this. Still, experience with such formalizations shows that it can be done in a straightforward (if tedious) manner.

In any case, one does not have to add axioms to $\mathsf{ZF}$ to study the Euclidean or non-Euclidean case. The axioms of Euclidean or non-Euclidean geometry simply pick out subclasses of the class of manifolds. Once you pick a formalization of "manifold" in terms of sets, it is a consequence of $\mathsf{ZF}$ that there are many examples of both types of manifold.

It is true that some things are not decided by $\mathsf{ZF}$, such as the Continuum Hypothesis. However, the statements "there are Euclidean manifolds" and "there are non-Euclidean manifolds" are both simply theorems of $\mathsf{ZF}$. Note that in the context of $\mathsf{ZF}$ it does not make sense to ask whether the universe itself is Euclidean, because the universe of sets is not a manifold at all.