Re: infinity

David Kastrup said:
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
>
> > I have already explained this over and over, but if you refuse to
> > acknowledge the logic behind the axiom of induction,
>
> It is irrelevant.
>
> > there is very little I can do to help you understand. Logical
> > implication is transitive, and as the below-referenced paper
> > explains, that is the logical foundation for inductive proof:
> > transitivity over an infinite chain of logical implications.
>
> Irrelevant.
>
> >> >>>I'm sorry, but if induction covers the entire infinite set it
> >> >>>does so with an infinite number of recursive iterations, and
> >> >>>that must be taken into account.
>
> Irrelevant.
>
> >> That much is incorrect; you are evidently thinking of induction
> >> proofs of algebraic formulas, and not of the sort I presented. I
> >> imagine that those are the only induction proofs you've
> >> encountered.
>
> > Incorrect. I am well-enough acquainted with inductive proof to point
> > out the reason why it works.
>
> Irrelevant. You are obviously _not_ acquainted at all with _any_
> method of proof if you fantasize that there must be some reason behind
> axioms.
>
> > Saying "the axiom says so" is not an explanation of anything except
> > your own inability to examine your working assumptions.
>
> It is not an explanation, it is the _validation_ of the proof. The
> proof is finished once you have reduced it to the axioms. Whether
> those axioms have reasons of themselves or are random (but consistent)
> nonsense is completely irrelevant.
>
> >> So, if I get you correctly, you're saying that induction fails to
> >> apply to F because 0 is not finite? Otherwise, you're saying that
> >> you don't get the solution to this elementary exercise.
>
> > I am saying that the infinite series of 1/2^n for n=1 to oo has a
> > definite answer of 1, and yet the proof states that it is always
> > less than 1 even for infinite n.
>
> The proof can state no such thing since inductive proofs hold only for
> finite n (of which there are infinitely many).
>
> Anyway, the proof is about (finite) partial sums of the series, and
> the limit is not such a partial sum, anyhow.
>
> >> If you do not accept the induction axiom as written, you do not
> >> accept the Peano axioms.
>
> > I accept the peano axioms as far as they go.
>
> Fine. Then you also have to accept induction and refute infinite
> members in th naturals.
>
> > There is some improvement that can be done, and there are caveats
> > with iductive proof as there are in all areas of math.
>
> There are no "caveats" with axioms. That's their whole point.
>
> >> You are not required to accept the Peano axioms, but if you're
> >> going to claim anything wrt the natural numbers, it would be to
> >> your advantage at least to understand what they say.
>
> > It would be to your advantage to understand WHY they say what they
> > say.
>
> Irrelevant for proofs.
>
> >> Your refusal to show how F fails to meet the hypotheses of the
> >> induction axiom, and more pointedly, your inability to see how that
> >> really *is* the issue, merely point to shortcomings of your own
> >> background.
>
> > Your refusal to examine the logic behind the axioms you use
>
> It is irrelevant for the math. You don't need to look behind axioms.
> They are valid without looking further. That's the whole point of
> them.
This is precisely the root problem with mathematics today. This Cantorian
nonsense is the fruit of an endeavor where it is against the rules to question
the rules. If you don't understand why the rules exist, you have no business
obeying them, unless of course it is just to get brownie points with your
academic buddies and get a good mark on someone else's test. Would that
mathematicians actually cared about what numbers mean, instead of just securing
their position in academia so they can spew nonsense and get paid for it. This
is why there's no Nobel Proze in Mathematics. Alas!
>
>

--
Smiles,

Tony
.

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