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I was reading The Joy of Cats, and on pg. 383 it goes:

Also Top is definable by topological axioms in Spa(F). However, a proper class of such axioms is needed.

Some random factoid, in context (pg. 219), also depends on how many measurable cardinals are assumed:

Prove that the following statements are equivalent:

  1. Set has a small codense subcategory,
  2. [T]here do not exist arbitrarily large measurable cardinals [equiv. there is not a proper class of them]; i.e., for some n, every ultrafilter closed under n-meets is closed under all meets.

So I suppose that we need not a proper class of axioms for measurable cardinals as such, although if we work with an axiom scheme of measurability, I guess we could have an arbitrary number of such axioms.

Anyway, how many axioms is too many? If the number of axioms not only equals the number of theorems, but even exceeds the possibilityX of more theorems than axioms as such, is that "too many"?

XHamkins lists, "Every proper class is bijective with Ord," as equivalent to AC under NGB. This indicates that it is possible to conceive of multiple "sizes" of absolute infinity, if we hold fast to, "A proper class is absolutely infinite." Or else proper classes wouldn't count as absolutely infinite, either (if they counted as having a "size" as such). At any rate, if X is a proper class of axioms and Y is a proper class of theorems, yet if global AC is dismissed (modulo NGB), would we have a possible discrepancy between the amount of axioms and the amount of theorems in play?

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  • Since there are proper-class-many FOL formulas externally, the axiom scheme is a proper class independent of large-cardinal height. However, statements like “There're not arbitrarily large measurable cardinals” really do change truth value as you climb the large-cardinal hierarchy. There's no set-sized axioms pinning this down uniformly reflecting the fact that the mathematics genuinely depends on unbounded set-theoretic strength. A proper class of axioms can have a very “small” deductive closure. And w/o global AC, there's no notion of ‘size’ for proper classes, or collapsing to Ord w/ it... Commented Jan 4 at 6:12
  • On the face of it, one can simply construct from A, B, C, ... theorems A, A∧B, B∧C, A∧B∧C, ... . Cardinality of that class should be equal to that of the "power class" 2^{A_i}. I'm far too thin on this and should read the archives though Commented Jan 4 at 16:01
  • Not sure if ∧ can be applied I-times for I very large. Cheap shot: axioms are theorems!!1 Commented Jan 4 at 16:02
  • @DoubleKnot what do you mean by “there are proper-class-many FOL formulas externally”? There are at most countably many variables, predicates, connectives, quantifiers, etc. from which to choose, and clearly every formula is finite in length/complexity, so I’m not sure what you’re saying. Commented Jan 6 at 13:09
  • @PW_246 This is one of those places where “externally” is doing real work ranging over all FO formulas across all language signatures/parameters/contexts, not just one fixed language, and you're talking "internally". For example, in set theory we constantly consider formulas with parameters φ(x,a1,…,an) where the ai range over all sets. The axiom scheme ranges over all formulas with parameters, not any set of them. This is why axiom schemes are not sets externally and ZFC proves only countably many theorems as you reasoned... Commented Jan 7 at 6:26

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You might consider a number of axioms to be excessive if all of the statements derivable from them could have been derived from a given proper subset of them.

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    @controlgroup Please note that it took a long time to prove that the axiom of parallels is independent from the other axioms of Euclidean geometry. The proof was by constructing non-Euclidean geometries which violate the axiom of parallels. Commented Jan 4 at 0:47
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    @controlgroup a half-baked version of that was my motivation for including the words 'of interest' in an earlier version of my answer. I was wondering, for example, whether you could prove anything with, say, just two of Euclid's axioms, and if so, and if they were the only things you were interested in, the other axioms would be superfluous . Commented Jan 4 at 7:49
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    This isn't a sufficient definition I think - if X0, X1... are statements, consider the axioms X0, X0^X1, X0^X1^X2... Commented Jan 4 at 15:28
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    Any proper subset unbounded above is sufficient to derive the full set, any finite set of statements can be safely removed so it meets your criterion, but you can't get smaller than the countably infinite subset, even by choosing different axioms (as each axiom can only refer to a finite number of the X_i). Commented Jan 4 at 15:31
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    Both axiom schemas (specification and replacement) in ZFC have similar properties, allowing you to safely remove any finite subset. Commented Jan 4 at 15:34
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An infinite number

Godel’s incompleteness theorems tell us that it is impossible to describe all mathematics with a finite set of axioms; therefore an infinite number is needed. Of course your infinite set has to be self-consistent, which is tricky. It also has to be infinite, which is also tricky.

A parsimonious set

Assuming you are ok with incompleteness, which you have to be because that’s what we’ve got, then you want the smallest self-consistent set that does the mathematics you are interested in.

For example, to do Euclidean geometry you need 5 postulates (plus some unstated assumptions that should probably be postulates). Take away the parallel postulate and you can do non-Euclidean geometry. Different axioms let you do different geometry. But, geometric axioms won’t let you do arithmetic; at least, not all arithmetic. For that you need different axioms.

Now, whatever maths you are trying to do, the axioms should be the smallest set that allows you to do that where none of them can be proved from the others; that is, all of them are necessary for the relevant maths.

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    @ScottRowe - The Continuum Hypothesis is one. It's proven to be independent of the ZFC set theory axioms, so you're now free to adopt it (or not) as extra axiom. But generally that's not done, because it's just too specific. You want axioms to be kind of general. You want to be able to do things with them. Commented Jan 4 at 15:56
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    @ScottRowe - A funny one is the axiom of Foundation (that says that every non-empty set has an element that has no elements in common with the set). Seems pretty innocuous, right? It rules out sth like x = {x}. It's independent of the other ZF axioms and can be dropped. Replacing it with Anti-Foundation apparently is useful to set up theories of non-well-founded sets which apparently are useful e.g. in computer science (for infinite streams, cyclic graphs). (Simply dropping Foundation will also leave large swaths of math intact.) Commented Jan 4 at 16:07
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    @mudskipper so when I see things like this in math which can be accepted or not, and which can have significant effects or maybe no effect, it makes me wonder about people who say that there is a Platonic reality to math, when to me it seems to devolve in to a forest of gibberish. Maybe I just don't know enough about math, but it's certainly not an encouraging area to explore. Commented Jan 4 at 20:43
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    @mudskipper From my own experience I'd say that 90% is a massive underestimate, at least if you're talking pure math specifically rather than applied. Mathematicians all have examples in their pocket to haul out when someone asks "what's the point of it", times when pure math research has turned out to have useful applications decades later (ask any mathematician and you'll hear "cryptography came from number theory" before you even finish your question), but in practice, from what I've seen almost no one in pure math cares about utility, it's essentially all l'art pour l'art. :P Commented Jan 5 at 14:54
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    @DaleM : You’re referring to the “unreasonable usefulness of mathematics” argument. I actually don’t find it that mysterious. If you want to know “how long do I need to burn the engine to reach the moon”, this is an inherently mathematical question (“how long” and “location of the moon”) Commented Jan 6 at 23:36
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I don't think it's the number of axioms that matter as much as the challenge of collectively arguing the validity of said axioms in a practical context. In a very informal sense, more axioms lead to an increased likelihood that the resulting conclusions cannot be applied in a useful way.

I say that informally because its a tricky thing to pen formally. As a trivial argument, any finite number of axioms can be combined into one axiom with a liberal application of conjunctions!

The story does get more nuanced when you start talking about infinite sets of axioms (or worse: proper classes!). Infinite sets of axioms can get squirrelly quick. The distinction between first order logic and second order logic is an excellent example of what sort of peculiarities can form.

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You have too many axioms if any axiom can be derived from the other axioms.

I believe this is equivalent to saying: "if all of the statements derivable from them could have been derived from a given proper subset of them" (as per another answer).

The reason being: A statement is derivable from a set of axioms if it can be derived from that set combined with zero or more other statements derived from that set (e.g. you derive A from S, you derive B from S+A, so B can be derived from S). So all statements derivable from a set is the same as all statements derivable from that set plus a statement derived from that set. And one might say an axiom can be derived from itself as a tautology, so if all statements are derivable from a proper subset of the axioms, then any excluded axiom must be derivable from the other axioms.

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There is no such thing as "too many axioms". Axioms are the formulae over a language which are assumed; a proof system then gives us the deductive closure of those formulae. Thus, who cares what set of axioms we chose initially? What is of interest is the theory generated by their deductive closure; which may have multiple distinct axiomatizations.

If there were a theory which could only be axiomatized with 1,000 distinct axioms, so what? There are doubtless complex theories that require a lot of specification. Being complicated is not a demerit of a theory at all. In the realm of all theories which exist, there are going to be theories which require many axioms. So be it. There is no normative question of how many axioms a theory should have. It is simply a matter of fact that this collection of axioms generates this theory; and we can study that theory.

There are infinitely many theories. All are "worth" studying.

By the way, I just thought of something to add to this.

One of the most "mainstream" sets of axioms out there is ZFC - it's taken as the "standard" foundation for mathematics.

Do you know how many axioms it has? Hint: it's not 8 or 9 or however you want to count them.

ZFC has countably infinite axioms! The axiom schemas of restricted comprehension and of replacement give us an axiom for each formula in the language!

It's interesting to consider that people may not even acknowledge that ZFC has infinitely many axioms.

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    But we haven't got infinite time, so a few brief theories is definitely the way to go. Another word for axiom is assumption, and you know what happens when we ass/u/me. Commented Jan 6 at 12:44
  • "What is of interest is the theory generated by their deductive closure; which may have multiple distinct axiomatizations." Is this a claim inherent in this that the same deductive conclusions can be reached through different, but similar ontologies? Commented Jan 6 at 23:02
  • @JD I'm not sure, to be honest my answer is very mainstream model theory / proof theory, not a philosophical analysis. It's an interesting idea though. But all I'm saying is that there can be different sets of axioms that are logically equivalent. Commented Jan 9 at 4:34
  • @JD There's so much to learn / research / think about, and so little time, but one thing your comment reminds me of is IEML by Pierre Levy - it is a formal ontology language based on a certain collection of ontological primitives. But long ago when I encountered it, I had the sneaking suspicion that the choice of ontological primitives was arbitrary. What matters is that you can build the same universe of concepts, even starting from a different chosen "basis". Concepts emerge by contrast from other concepts. I think this is similar to what you're saying. Commented Jan 9 at 4:43
  • @JD But on the other hand... we might say the ontology is better given by the signature - this is where we state "what things exist". So we might have 2 equivalent sets of axioms but over the same signature. Or we might have 2 equivalent theories in totally different languages. And I think that's called "bi-interpretable". So yes, I would say... we can have interesting insights about different "ontologies" expressing the same "ultimate thing". Like, when I have the time I want to study NGB set theory, since you get a "class" of "all sets". Seems... ontologically intriguing. Commented Jan 9 at 5:04

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