someguy1 » October 24th, 2015, 8:54 pm wrote:Natural ChemE » October 24th, 2015, 7:13 pm wrote:someguy1,

I feel that I understand this subject a lot better now. Thank you for your time in explaining it to me!

Oh you are very welcome. Thanks for raising so many stimulating issues. I hope you're not done! We've just cleared out the underbrush, the interesting stuff is ahead.

No matter, if anything occurs to you feel free to ask. You had such a huge rush of wild interconnected ideas ... did I manage to address most of it at some level?

Yeah, I definitely feel like a lot of my confusion has been cleared up as far as it can be.

For example, the problem with my bijection was that it excluded non-terminating decimals, right? To me, numbers are constructions, and non-terminating decimals are just a type of construction that is itself non-terminating.

I see the idea of the set of reals being non-countable as a direct consequence of the implicit construction approach that separates a number's construction from a set's construction. This is, my construction for the set of all reals ultimately constructs all terminating decimals that the construction for a numeric expression of ever would. However, my set of all reals, as defined by my construction of it, would never include a non-terminating decimal in the same sense that no construction of would ever fully reproduce it in decimal form, either.

Due to this perspective, I see the non-countablity of the set of reals as a direct consequence of the implicit, unacknowledged (and thus ambiguous) decision that the construction of a set must occur only after the construction of the members to be added to that set. This makes sense when folks see numbers and sets as fundamentally distinct; a constructionist like me sees both as merely being constructions themselves, making merged constructions possible.

While the full point is above, I would say this in another way. This is, I see cardinality as sort of a loop process in modern programming, wherein the order of cardinality is directly equivalent to the depth of nested loops that must go arbitrarily far. For example, any set is countable if and only if it can be constructed in code as

- Code: Select all
`List<TypeOfSetElements> countableSet = new List<TypeOfSetElements>();`

for (int i=0; i -> infinity; ++i)

{

countableSet.Add( FiniteProcedureForForNewElement(i) );

}

`FiniteProcedureForForNewElement(i)`

; rather, some elements (such as the non-terminating decimals you pointed out) must be generated by infinite procedures `InfiniteProcedureForForNewElement(i)`

. As such, the full construction method for the set of reals would have to contain at least one more nested infinite loop to construct the reals.Assuming such a construction procedure to be like

- Code: Select all
`public TypeOfSetElements InfiniteProcedureForForNewElement(int i)`

{

TypeOfSetElements toReturn = new TypeOfSetElements();

for (int j=0; j -> infinity; ++j)

{

// Omitted here is some procedure for

// constructing a real number.

// This omitted procedure may itself contain

// nested infinite loops.

// If this omitted procedure does contain an

// infinite loop, then the Continuum Hypothesis,

// which states

// "There is no set whose cardinality is strictly

// between that of the integers and the real

// numbers."

// is wrong because other sets may be constructed

// without a nested infinite loop. However, if this

// construction doesn't contain an infinite loop,

// then there's no cardinality between that of

// integers and real numbers, proving the

// Continuum Hypothesis to be correct.

}

return toReturn;

}

However, I'm entirely comfortable with the fact that my construction of all real numbers must itself run for infinity to construct any construction that must itself be constructed by running to infinity, e.g. non-terminating decimals. This directly leads me to see the differing cardinalities of the sets of integers and reals to be an immediate consequence of the common decision to construct non-terminating decimals in a nested loop. However, since I can (and have) demonstrated a construction that fully performs the same by merely flattening the loop, this notion of cardanlity is nothing to me but a direct, immediate, and obvious consequence of the classical distinction between numbers and sets of those numbers.

And this is why I'm satisfied to leave it here. I feel that the fundamental issue was that classical mathematicians see there to be a reason to separate the construction of numbers from the construction of a set of those numbers such that cardinalities make sense. And because I'm disinterested in false distinctions between these constructions, I'm entirely content to consider this resolved.

Anyway, the bottom line is that I really do appreciate you explaining this subject to me. I now understand that the distinction between numbers and sets of them to be the fundamental assumption that I was missing, and all the stuff that I've been reading now makes sense within that context.