If it's just a cognitive illusion, why is it so hard to find contradictions in axiom systems about integers that were generated by humans intuitively, like PA?
We seem to have a superpower for inventing apparently consistent axiom systems that make "intuitive sense" to us. The fact that we have this superpower is even machine-checkable to a certain degree (say, generate all proofs of up to a million symbols and look for contradictions), but the superpower itself resists attempts at formalization.
We work in axiom systems that have not been proven inconsistent because in the past we have reacted to inconsistencies (such as Russel's paradox) by abandoning the axioms. I wouldn't call this a superpower but a bias.
Russel's paradox is a cliche but it's really illustrative. The principle of noncontradiction says that, since we have arrived at a false statement (the barber shaves himself and doesn't shave himself) some of our premises must be wrong. Apparently the incorrect premise is: given a property P, there is a set of elements that satisfy P. What...
Followup to: What's a "natural number"?
While thinking about how to make machines understand the concept of "integers", I accidentally derived a tiny little math result that I haven't seen before. Not sure if it'll be helpful to anyone, but here goes:
You're allowed to invent an arbitrary scheme for encoding integers as strings of bits. Whatever encoding you invent, I can give you an infinite input stream of bits that will make your decoder hang and never give a definite answer like "yes, this is an integer with such-and-such value" or "no, this isn't a valid encoding of any integer".
To clarify, let's work through an example. Consider an unary encoding: 0 is 0, 1 is 10, 2 is 110, 3 is 1110, etc. In this case, if we feed the decoder an infinite sequence of 1's, it will remain forever undecided as to the integer's value. The result says we can find such pathological inputs for any other encoding system, not just unary.
The proof is obvious. (If it isn't obvious to you, work it out!) But it seems to strike at the heart of the issue why we can't naively explain to computers what a "standard integer" is, what a "terminating computation" is, etc. Namely, if you try to define an integer as some observable interface (get first bit, get last bit, get CRC, etc.), then you inevitably invite some "nonstandard integers" into your system.
This idea must be already well-known and have some standard name, any pointers would be welcome!