How do skeptics explain axioms not being arbitrary?

Question

I get infinite regress but surely the axioms of ZFC or arithmetic were not so much chosen as discovered and intuited and thought about. They certainly didn't just grab whatever was around them and say that it doesn't natter at all what you say your axioms are.

I get that infinite regress makes all basic beliefs doubtful, but surely, some truths are better than other truths.

Also, infinite regress itself seems to be a system of concepts on its own which needs to be composed. It also makes a lot of odd assumptions itself.

But the most damning conclusion is that all knowledge is arbitrary and that we cannot know anything. Sometimes even the meaning of these words themselves. This is nonsense. Reasoning is not to be doubted whatsoever. Yet, people still do it.

For instance, they attack notions like undefined terms in set theory or type theory. They say set type 0 is undefined and therefore means nothing. But yet, clearly we mean something by it. We would not say it is not a type. We would not say it is a set. Etc. It is undefined because there are no other things to define it. A set is defined as a collection of things. To truly try to define everything would also mean defining nothing.

They say how do we know what a set is if it is undefined?

I respond saying these are not arbitrary.

Are there any global skeptics who believe in absolutely nothing, not even semantics and reasoning that can salvage their position of non-position?

Axioms are not arbitrary, although they are assumptions.

Edit: I am not entirely sure if you can even truly think of nothing. To truly have no assumptions is not possible. You must have some facts.

Answer 1

There are indeed radical sceptics and there have been in history. In particular, there are sceptics like Sextus Empiricus who even wrote a treatise called "Against the Logicians", attacking all logical reasoning. Pyrrhonistic scepsis has generally gotten a bad rap because many philosophers seem to think that it is making assertions or knowledge claims, or that it is somehow inconsistent or untenable. But this is based on a profound, almost willfull misrepresentation and misinterpretation. To reason or to apply arguments, one doesn't need to "believe in reasoning" (whatever that means). Just like walking - in order to walk, I don't need to "believe in walking". I just do it. It's an activity. The sceptic can engage in philosophy in order to challenge other's beliefs, or for whatever other reason; as sceptic they don't need to believe in anything specific to do so, and anything is really up for discussion - especially any "everyday, common beliefs" people may have (if they have any).

For a sympathethic presentation of radical scepticism, see Arne Naess' book "Scepticism".

Answer 2

Radical skepticism is not about beliefs being meaningless or random, but rather beliefs being incapable of being moved to a state of knowledge. From WP:

Radical skepticism (or radical scepticism in British English) is the philosophical position that knowledge is most likely impossible.1 Radical skeptics hold that doubt exists as to the veracity of every belief and that certainty is therefore never justified.

That is to say, axioms, which are systems that present meaning, are certainly not random, because language and meaning itself is not random. Words are used in such a way that you simply cannot present them in random order. And so axioms must follow this, since axioms help to organize language into meaningful units that relate in a particular logical way. But this doesn't in anyway undermine radical skepticism.

Radical skepticism is not about the failure of logic or meaning, but about the failure of justification. From WP:

Justification (also called epistemic justification) is the property of belief that qualifies it as knowledge rather than mere opinion. Epistemology is the study of reasons that someone holds a rationally admissible belief (although the term is also sometimes applied to other propositional attitudes such as doubt).

Thus, a radical skeptic objects to the idea that there is anything that resembles justified, true belief which can used as a judgement to determine that what is believed is in some sense certain. Consider that one can have an elegant set of first claims about the physical world, can have a wonderful set of logic, and draw with logical certainty some inferences, but then cast away that entire apparatus with the argument that one is suffering from an extended delusion like Walter Mitty. Mitty was a character who believed, and reasoned, and justified, but his fantasies were just plain false. And false belief is not knowledge.

Thus, in this scenario, the non-accidental nature of axioms is irrelevant, because justification is rejected on the basis of the falsity of belief, not their semantic or logical incoherence. And so it is for the radical skeptic. Given any set of strong axioms that appear to be a foundationalist basis of knowledge, the skeptic finds away to defeat the argument that adequate justification is to be found of taking the belief in those axioms to a state of justified and true belief.

Answer 3

Two things:

1. There is something haphazard about the assortment of the ZFC axioms (c.f. Kanamori[03] and Maddy[88]). This has been acknowledged especially in light of the "quest for new axioms" that are either alternatives to, or unifiers of, the ZFC menagerie. But so it is understood that the axioms are not as unified as might be desirable, "by themselves" that is (and perhaps an inaccessible cardinal is the minimal noncircular solution to this problem?^W). Moreover, then, plenty of alternatives have been considered in much detail, and the prevailing attitude towards the even-higher menagerie is tolerance (AKA pluralism, or "the" set-theoretic multiverse).

2. The SEP entry on type theory, in a section about the relationship with set theory, says:

It is clear intuitively how we can explain type theory in set theory: a type is simply interpreted as a set, and function types A → B can be explained using the set theoretic notion of function (as a functional relation, i.e. a set of pairs of elements). The type A → o corresponds to the powerset operation.

The other direction is more interesting. How can we explain the notion of sets in terms of types? There is an elegant solution, due to A. Miquel, which complements previous works by P. Aczel (1978) and which has also the advantage of explaining non necessarily well-founded sets a la Finsler. One simply interprets a set as a pointed graph (where the arrow in the graph represents the membership relation).

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^WThe minimal technical solution is a worldly cardinal, that will happen to have countable cofinality; but insofar as worldly cardinals are defined in overt reference to ZFC, indeed defined as that technical solution, then they are not as appealing as inaccessibles, here ("Worldy cardinals are justified because they solve the problem," is trivial, because it can be translated as, "Solutions to this problem are justified because they solve the problem," whereas it is less trivial to say, "Inaccessibles are justified because they solve the problem" (note that inaccessibles do have the worldliness property, but it is not their key identifying feature)).