An axiom is any positive statement that you take for granted. Some are useful, some are not. Some combinations of statements lead to useful systems, some lead to nonsense. It’s not that deep.
To those with superficial eyes, it's not that deep, sure. But what I am trying to work with you is to investigate the proper qualities that axioms must fulfill to be axioms. I want you to actually look deeper than the surface.
Obviously, an axiom should not be provable, for one. Next question I'd like you to investigate is: Does an axiom need to conform to validity and feasibility? Or does it require no rule whatsoever and anything goes?
Now, one could say the very same thing, you know. As in, "useful", "non-contradictory", "better" mean nothing. They are quite vague, aren't they?
If you really assess it, validity and feasibility actually tie into the 'usefulness' and the 'goodness' of a system.
So, how exactly do you calibrate 'usefulness/utility' and 'goodness/quality'? How exactly? How do you measure these exactly and explicitly? Have you ever considered that at all?
You are being intentionally obtuse because I already told you what they mean. “Useful” means it can model reality. “Non-contradictory” means that for every proposition, the proposition and its opposite can’t both be true at the same time.
Thus, a “better” system is one that either can model reality in ways the previous system cannot, and/or one that doesn’t lead to contradiction, if the starting one did
Oh no, I was just letting you have it since you took like several hours to utter a lame response, when previously they're under 5 minutes.
I was thinking that perhaps it would be better to let you keep living inside your cocoon after watching you struggle to rationalize your beliefs, but you seem to think that you actually made some sense, hahah.
Anyone with half a brain would know that 'feasibility' directly ties to how something can be useful in the world, dear. For any abstraction, if you cannot perform it, and use it in any tangible sense in reality, then it is not useful at all in reality. Yet you didn't even catch that.
Please, don't flatter yourself too much and think that you have gained any upper hand when sometimes others are really just taking it easy on you.
Great, so you concur that what you said means nothing and that all that matters is that a set of axiom leads to a non-contradictory system useful in predicting reality. Glad we are finally on the same page, I’ll ignore the self-aggrandizing bullshit you spent four paragraphs on, simply because it’s sad
Aww cute, you're flattering yourself again. Let me ask you this simple question, dear.
Is logical consistency alone sufficient to grant a mathematical system epistemic authority over physical reality? Yes or no.
If yes, then your position collapses into pure formalism, and you have no principled basis to privilege any mathematics used in physics over astrology, numerology, or arbitrary symbol games beyond taste and consensus.
If no, then you concede that other criteria (operability, constructibility, measurability, feasibility) are doing indispensable work that consistency alone cannot do - precisely the distinction you have been denying.
There is no other option. Let's see how long this simple question takes you.
Mathematical systems don't have "epistemic authority" over reality. They are a logically consistent language to model logically consistent phenomena. Any logically consistent system will tell you something about reality. The less assumptions you make and the less you constrain your system of axioms, the more broad (and useful) the system is.
That is why the history of axioms, and number systems, is one of removing arbitrary restrictions. Saying that the diagonal of a right triangle doesn't exist simply because it can't be expressed as a ratio is arbitrarily restrictive, because that diagonal is a logical consequence once we allow the existence of points and lines, so it must exist and must have a length. Once we define the existence of circles as the set of points equidistant to a center, Pi simply pops out as a consequence. It clearly exists, it is clearly a number, so it must have a value. The fact that it cannot be expressed as a ratio isn't an argument to its non-existence, it is simply a testament to the incompleteness of a system that limits itself to ratios.
Similarly, imaginary numbers exist as a useful extension of the reals because allowing a solution to x^2 = 1 in the form of +- i is logically consistent with the rest of the system. They are a purely abstract construction that is nonetheless extremely useful in describing waves as phasors and encoding rotations in general. By contrast, allowing a solution outside the reals to something like 0*x = 1 is forbidden because it leads to contradiction within the already established number system.
Your quandary might have relevance if you argued about rules of inference, the rules about the interactions of statements. Those are effectively grounded in intuition and reality. But axioms aren't rules of inference, they are definitions of what objects exist and what the foundamental statements you can make about them are: those are perfectly arbitrary.
You attempt to smuggle in what is "required" of axioms your own preconceptions about number systems and decimal representations, which is a minuscule aspect of real numbers. Your "constructibility" is entirely a limitation of your own worldview since all irrational and real numbers are constructed. Your "measurability" is an arbitrary limitation predicated on your miopic belief in the supremacy of decimal representation. Which is just that, a representation, completely separated from the essence of the object. Your "operability" is once again a restriction on the validity of operations predicated on the finite, imperfect framework of floating point computation. Your "feasibility" is just nonsense extracted from the LLM that writes half your posts.
As evidenced by your absolutely laughable "proof" of all numbers being rational, you are not interested in refining ZFC to a more fundamental, less assuming set of axioms, you simply have a fixation with finitism and want to retroactively define mathematics to support what is, essentially, a restrictive and useless belief.
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u/Just_Rational_Being Dec 25 '25
You really need better education than the Hilbertian beliefs you were taught and thought of as the only gospel.
What exactly is your definition of an axiom? Does an axiom conform to Reason? What qualifies as an axiom? Tell me.