HP PRIME not raising overflow/underflow may cause critical errors

[robve]

(03-20-2021 06:05 AM)Han Wrote:  Context makes a huge difference in mathematics, though. And depending on the context, it may be 1, or 0, or undefined. It is defined in those contexts by choice in order to have a consistent system. Hence the "consensus." The wikipedia link gives plenty of well-reasoned examples of why they can be taken as any of these. Here is another resource:

LOL, I do not see any "consensus" in the resources you point to that \( 0^0=1 \) is a universal truth.

Perhaps you are an adherent of a branch of "constructive algebra" that allows you to make up your own rules as you go as long as the algebraic system you're working with is sound. That is perfectly fine. But do not impose your belief system on others when it is universally incorrect. That's the same as saying \( \pi=3 \). It may hold in your custom algebraic system (your "context"), but do not tell everyone that \( \pi=3 \) when they actually want to measure the circumference of a circle in Euclidian space.

(03-20-2021 06:05 AM)Han Wrote:  EDIT: Also, your explanation is in fact not mathematically sound. For \( 0^n \) your assumption is that \( n \) is a positive integer.

Where do I state the condition that n is integer or x has constraints? You are saying that zero cannot be raised by a non-integer n and therefore the "trail of zeros" cannot have a "fraction" of zero? There are no constraints due to universality, i.e. constraints impose exclusions to create a custom algebraic system.

So thank you for actually helping to make my point that there are no constraints on n and x. Hence, the expression I've used is to illustrate in a very simple way that without constraints we both have \( 0^0=1 \) and \( 0^0=0 \), which is inconsistent. Therefore, \( 0^0 \) is undefined. An algebraic system falls apart when counter examples demonstrate that the axioms and rules of the system are inconsistent.

On this highlight, the great mathematician Hilbert was heavily criticized. It is fascinating history.

For all his successes, the nature of his proof created more trouble than Hilbert could have imagined. Although Kronecker had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea" — in other words (to quote Reid): "Through a proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page) was "the object". https://en.wikipedia.org/wiki/David_Hilbert

Custom algebraic systems do exists. You can create any such system as you like, including infinitely many custom algebraic systems in which \( 0^0=1 \) as long as you prove that the system is sound. But the point is that these systems are customized to allow \( 0^0=1 \) to be sound. This deviates from our universal system of arithmetic, i.e. such custom algebraic system is not isomorphic to our universal system of arithmetic.

If you claim that \( 0^0=1 \) then it is up to you to prove that it is universally true in our common system of arithmetic. I'm eagerly awaiting...

- Rob