Re: universal quantification, because g⤨(g⁻¹(x)) = g(y) [1/2] Re: how

On 05/18/2024 11:16 AM, Jim Burns wrote:

On 5/18/2024 12:09 PM, Ross Finlayson wrote:

On 05/16/2024 09:50 AM, Jim Burns wrote:

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[...]

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I think that "correct", in context, is the entire
context, which is exactly what deductive inference
contains, explaining when inductive inference either
must complete, or meets its juxtaposition, with
regards to any two forces that balance and align
in symmetry.
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So, what you are claiming is that inductive inference
is invincibly ignorant,

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I am claiming that inductive inference
is invincibly modest.
Post.inference, we only assert claims about
whatever.it.is we described pre.inference.
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Perhaps that doesn't seem modest,
because whatever.it.is is infinitely.many,
but induction holds for infinitely.many cisfinite ordinals
in the same way that geometry holds for infinitely.many
right triangles. Completely.
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A given schema for induction has no more correctness,
in its own vacuum, than any other,

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Induction on the cisfinite ordinals
⎛ those countable.back.to.0 after only
⎜ those countable.back.to.0
⎝ and also 0
is a theorem.
Theorems are not optional.
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and when they're put together and don't
agree, then either they don't, and don't, or
don't, and do.
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"Not.ultimately.untrue", ....

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Induction on the cisfinite ordinals
is not.first.false in a finite sequence of
only not.first.false claims
which begins "A cisfinite ordinal is ... ".
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One can contrive simple inductive arguments
that _nothing_ is so.

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An example of such an argument would be clarifying here.
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So, I'd say your definition of "correct", isn't,
and is simply a declaration of "relative" and "blind".
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No offense meant, of course, it's so that paradoxes
are to be resolved, not obviated.

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Says nothing, says nothing, says nothing, says nothing, ....
See, just saying so doesn't make it so,
something that _goes_ has a _place_
to go.
Consider for example the mathematical limit,
when, and if, all the terms are related there's
only one rule need follow, and an arbitrary
competing rule, shares no terms, so is altogether
not relevant, we can say that the limit exists
and we can get close enough to establish
vanishing differences.
Yet, the "infinite limit" is already stronger,
close enough isn't good enough, when competing
conditions would otherwise result its nullity,
in effect.
Then, the "continuum limit" is an idea of
even a strong sort of setting, and about
strong enough for purpose of a continuous
milieu altogether, all relevant, all book-kept,
like the natural/unit equivalency function,
or most any other continuum limit you might
come across in all sorts of mathematical
treatments of probability and physics.
For mathematical terms to maintain a relevance
to each other, there's established as of the
"relephant", as it were of each other. "Relephants:
never forget".
In case of fire: break glass.