I suppose you would like me to paraphrase so you can judge my comprehension. — keystone
I incorrectly claimed that the S-B paths converged to a limit. — keystone
I think this may be an important point to you because you are stressing the importance of completeness to calculus. — keystone
What I'm suggesting is that by starting with uncountably infinite objects (corresponding to real numbers) you are effectively starting with the 'bottom of the tree'. And that agreeing to the former and not the latter is wanting your cake and having it too. — keystone
A continuum defined by numbers — keystone
[in k-musings] numbers defined by a continuum. The ordering of numbers in this [k-]system does not need to be complete. — keystone
Would you agree to either of the following?
1) A continuum is defined completely by numbers.
2) A line is made up entirely of points. — keystone
I don't think there's a need to define the limit of an algorithm. — keystone
Are my proposed algorithms that different from Cauchy sequences?
— keystone
Indeed they are! I EXPLAINED this. I don't understand what you don't understand in my explanation.
(1) An algorithm is finite. A Cauchy sequence is denumerable. And an equivalence class of Cauchy sequences has the uncountable cardinality of the set of equivalence classes of Cauchy sequences.*
* I think that sentence is right.
(2) There are only denumerably many algorithms, but uncountably many equivalence classes of Cauchy sequences.
(3) Cauchy sequences have a limit. But if we somehow defined the limit of an algorithm, then that would be infinitistic (unless some actual rigorous workaround could be formulated). — TonesInDeepFreeze
very number-centric view. — keystone
Your "line", the k-line, has NOTHING on it, as YOU said. So 'continuous' is not even applicable. And there is no infinite set of cuts on the k-line that comes after all the rows. You just now admitted that.
— TonesInDeepFreeze
Over and over you repeat the same point, as if I'm not understanding you. I understand what you're saying. — keystone
Then you added more apparatus that doesn't seem to me to improve the more basic and original goal that was not being addressed. Then you went further about "higher dimensions". I'm not sufficiently interested in whatever that's about to invest time and energy on it, while instead my curiosity is with the original questions of defining ordering and the operations. — TonesInDeepFreeze
So while the mathematician is still in the pre-formalized stage, deepening and extending the intuitions, she is putting herself into a kind of "intellectual debt". That is, the mathematician eventually is going to have to "pay" for the intuitive commitments with the hard cash of formalizing them. — TonesInDeepFreeze
So, in set theory, there is both the tree that doesn't have a final row or "row infinity" and the continuum. This is not having our cake and eating it too. Whatever we have comes from proofs from the axioms. The axioms are productive enough to proof the existence of many things including: the continuum, the S-B tree, finite algorithms, etc. — TonesInDeepFreeze
You are in an intuition stage. If you ever followed through to write some mathematics, then you would confront the debt you're accumulating and pay it off with rigorous formulations. But, in the meantime, one still needs discipline to not just mouth a bunch of incoherent mental picture stories. Even with intuitions, one would like not to commit to informal contradictions (unless one wants to base the proposal in a paraconsistent logic). Which is to say, crankery is a dead end. — TonesInDeepFreeze
You seem to have a notion that we have to distinguish numbers. No, every object is a set. — TonesInDeepFreeze
I don't know the purpose of this exercise. — TonesInDeepFreeze
I wanted to bounce my pre-formalized idea off of someone to see whether it was worth me investing in formalizing it. — keystone
the proofs come from the axiom— keystone
unless I can prove the axioms to be inconsistent there's no point discussing my musing. — keystone
While I would have deeply appreciated you trying to truly understand what I'm trying to say, I fully acknowledge that it is reasonable for you to not want to invest the time into it. — keystone
when discussing the standard position on an intuitive level many paradoxes arise. — keystone
The standard position doesn't gel with our intuitions — keystone
I didn't say that I'll only consider formalizations. I have been interested in the earlier proposals though not formalized. Rather, I said that I'm not inclined now to study your latest revisions. — TonesInDeepFreeze
Somehow, I don't believe you. To formalize you'd have to know what formalization IS. Be honest: Learning what goes into an axiomatic formulation is not a goal for you. — TonesInDeepFreeze
So, hopefully, you understand now that there's no "cake and eating it too" about the S-B tree and Cauchy sequences in set theory, or generally in set theory having both finite algorithms and infinite sets. — TonesInDeepFreeze
No, set theory shows how the paradoxes with the naive notion of sets are avoided. — TonesInDeepFreeze
Set theory is quite intuitive to me. I listed the axioms for you. I find each of them to be eminently intuitive. — TonesInDeepFreeze
huge investment of time and money — keystone
If ZFC is consistent then there's no cake and eating it too. — keystone
No, set theory shows how the paradoxes with the naive notion of sets are avoided.
— TonesInDeepFreeze
We've been down this road already. — keystone
Given that everything fits nicely together for you and in your view the paradoxes are addressed, I can see how you're not motivated to pursue a potential infinity solution. — keystone
Let's not debate my motivations. — keystone
Let's leave it at that. — keystone
Formalization is the way for an idea to be treated seriously. — keystone
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