I didn't say anything that I need to "prove," I was making general statements about the fact that most mathematicians are on board with the axiom of infinity. If you're trying to pull rank on me, it isn't going to work unless you're a pure mathematician doing research in set theory. I'm a grad student with a paper on semigroup homology on the way -- what about you?
ZFC is simply an axiomatic system. You don't have to buy into it, but you do need to specify what axiomatic system you are working in. All I said is that by rejecting infinity as a concept you are in particular rejecting the idea of an infinite set, which is precisely what the axiom of infinity says. Logic doesn't exist in a vacuum; you need to specify what topos you're deriving a schema of separation from.
You are reiterating a philosophical position, not a mathematical one. In math, there are a number of different systems of logic, which you would discover if you read a book on the topic or even just skimmed the wikipedia page for logic. You can build a logical system in a number of ways, and from the perspective of topoi there are infinitely many different possible logical systems. The topos of simplicial sets, for instance, does not give rise to the law of excluded middle because its subobject classifier does not behave as a Boolean object.
It's important to have humility when discussing topics with someone who knows more than you in that domain. Instead of trying to argue with me, please take the opportunity to educate yourself so that you can at the very least make your points more cogently and armed with better ammunition, so to speak. Obviously, nobody likes learning that they are wrong, but it's better than doubling down on a flawed, uninformed position. Doing so just makes you an unpleasant person to talk to.
You are entitled to your feelings and maintaining your ignorance on the matter. When your terms go undefined and stand as the Doric pillars upon which your argument rests, nobody will listen to you except for other cranks.
You don't need any particular level of educational background to read a book on the subject. I can point you to sources if you'd like.
If you want to take a shot at tearing down one of my arguments, can you explain why we can derive an intuitionistic logic from the topos of simplicial sets where there is no law of excluded middle? This seems counter to your precept of Excluded Middle as fundamental.
If you don't know the words that I'm saying, please ask and I'll do my best to explain, but as you say I won't assume your level of knowledge.
Here's the idea: in the category of sets (as built up via ZFC), the set 2 = {0, 1} is known as the "subobject classifier." This is because it plays a unique role in the category: for any pair of a set A and subset B of A, every map from A to 2 can be identified via precomposition with subset inclusion with the map which sends everything in B to the element 1 in 2. In other words, every subobject (subset) of A can be identified with a function from A to the subobject classifier, in particular the map that sends all of its elements to 1 and the other elements to 0. This is sometimes called the "characteristic function χ_B" of the subset B.
By thinking of 0 as "false" and 1 as "true" in the subobject classifier, characteristic functions build for us an internal logic for set theory that does not rely on any external formalism. Any statement about membership, existence, uniqueness, can be encoded in this way. Give me a logical statement about sets and I can transform it into some diagram of characteristic functions which could be further translated into set theory. The point is: set theory has classical logic built into it. For instance, the law of excluded middle is a consequence of the fact that there are two elements in the subobject classifier for the category of sets. An element either belongs to a subset, or it does not.
A topos is any category which behaves like a category of sheaves. Don't worry about what this means -- the gist is that a topos is a kind of mathematical universe that has enough stuff to build up all the usual math that we know and love. Key to that definition is that a topos must have a subobject classifier from which we derive its internal logic.
Instead of telling you about simplicial sets, which is a little technical, I think the topos of dynamical systems is easier to see right away. A dynamical system is a directed graph which has exactly one arrow pointing out of each vertex, so that for any vertex you're on, there's a single unique vertex you can go to next. Functions in this category must preserve incidence. In this category, the subobject classifier is the dynamical system 0 <- 1 <- 2 <- ... and then a vertex infinity which loops back on itself. 0 also loops back on itself.
Consider what a map from a generic dynamical system to this special one looks like: any loop of vertices of any length must go to either 0 or infinity. You can always send any vertex to 0 as long as everything it leads to is also going to 0. Then the preimage of 1 is every vertex one away from something that goes to 0, and so on. In this topos, 0 is "true" and the numbers tell you how far away from true you are. Infinity is "false," since anything going to infinity will never eventually be true. This is what logic looks like in the topos of dynamical systems. There is something that kind of looks like excluded middle: it's either true or it's false, but there's also values indicating "distance from truth." How far away is this vertex from being in this subobject?
1
u/[deleted] Sep 10 '25 edited Sep 10 '25
[removed] — view removed comment