Mathematical truths are not contingent upon sentient beings for their truth value.
Their existence is contingent upon those sentient beings, and also on the axioms chosen. Mathematical entities are concepts, which do not exist ontologically.
As a matter of fact, it is incorrect to “single out” the mathematical concepts from among the infinitely many concepts. All the concepts are the product of the mind, all are abstractions of reality. What they are “abstracted from” does exist (well, maybe!), even if there is no one to perceive them, but
the concepts do not.
Proofs that are contingent upon axioms are inductively derived.
Sheer nonsense. Inductive methods (which are never “proofs”) are based upon “basic
principles” which come from observation (like the principle of conservation or matter / energy / momentum / etc.). “Real” science uses induction. The basic principles are subject to change if and when it necomes necessary - that is when the predictions of the system are out of synch with the observed reality.
Deductive proofs are true implicitly and necessarily as a matter of logic not induction from axioms.
Are you joking here? Deductive methods are based upon axioms.
Axioms are arbitrarily chosen, and they are “true” by definition. The statements derived from the axioms are “defined” true
within that axiomatic system. The actually chosen axioms of mathematics are very useful when applying to reality, but they could be chosen differently.
You really need to learn the basics. With special emphasis of set theory, and mathematics.
The difficulty here is that a “null” world necessarily depends upon and keeps hidden by presumption, an entity, the “world,” or “container” in which the “null” obtains. So there is a necessary element even here. It may be different in kind from the distinct particular entities that are being counted, but it is a necessary aspect for the concept “possible” to even be possible.
You are obviously not a mathematician. You are in dire need to learn set theory. The concept of a “null-set” is not an “empty” container. There is no “container”.
One could not end up with a real world populated by any number of real entities that would obtain from a null world without something in that null world to produce the countable entities.
Totally out of topic. We are not concerned about the actual reality, and “how” it came to be. We are only concerned about abstract, possible worlds. If you do not like the “null-world” proof, because you cannot understand the math behind it, I gave two “actual” worlds, each with a different quark in it, and since the intersection of these worlds is “null”, there is no entity which would “appear” in all possible worlds - there is no “necessary” being. Q.E.D.