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andrewr

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I have been looking at the idea of 1:1 correspondence as a method of determining set size/cardinality, and have noticed that the principle allows for inductive proofs, which I think are properly constructed, that can come to conclusions which are clearly wrong under traditional set theory if used inconsistently; for integer sets are open -- (finite unbounded elements, having no supremium / therefore an open set) but not all sets are that way. Some sets CAN have supremiums.

For, it's not clear to me that 1:1 bijection can really prove something is infinite, rather than "finite unbounded" and I'd like someone to discuss the distinction, and how his tool of 1:1 correspondence really provides a valid and consistent tool that can be used in inductive proofs without fear of randomly confounding something that is really finite unbounded with something infinite.

eg: I'll show an inductive construction of the counting numbers, and then construct a proof based on the same principle that could potentially confound the two ideas (but doesn't have to) without making a mistake in the proof. (At least, I don't see a mistake...) and then could you explain to me exactly what kind of infinity Cantor had in mind for Aleph naught?

First off, consider what the set of counting numbers is:

The set of counting numbers, (A

eg: I would construct it like this for clarity.

N

The next step of the construction is always to add '1' to the last counting number that was united to the set.

N

N

N

N

And if the iterative process is allowed to go on forever, the set constructed is by definition the set of all counting numbers who's cardinality is therefore ℵ

A

When Cantor claimed this set size/cardinality to be the definition of the first infinity, I think he meant to say that the set contains (eg: *inside* the set) an infinite number of elements by axiom; and not by proof. And I am, of course, therefore not asking him to prove it -- but I am concerned that this has unintentional side effects -- for infinity is NOT the same as "finite but unbounded" in traditional set theory, eg: the individual elements of the set of counting numbers are each finite -- but by the same token (finite elements) I simply couldn't prove that the set is infinite inductively, but Cantor gives me an axiom with which I supposedly can always accurately measure when something is infinite; and exactly so -- not plus or minus one unmatched element; and I want to explore what that really means -- for I'm unsure that I truly understand it, or him.

For I'm pretty sure Cantor is claiming that he is measuring an *infinite* number of elements, whenever a set has a inductively provable 1:1 correspondence with counting numbers. He doesn't intend a size that is finite, even if we don't know how large.

For example; if I had all the decimal digit values of an irrational number that happens to have no zero digit places, and contained in a sequence for convenience, but not strictly necessary -- (eg: a sequence, is a set which in this example I'm making the successors be in strictly decreasing magnitude), and this set contains a supremium if there is one ... "0" ? but even if it doesn't it wouldn't affect the value -- for all nonzero values are in the set.

irr = { 3 .1 .04 .001 .0005 ... }

Then clearly that set would be infinite in length, for the value is irrational and the digit places summing up to the irrational value, by definition, can never end; Thus, such a sequence by even being listable is clearly in 1:1 correspondence with counting numbers; and the cardinality is aleph naught.

So, if I add up ALL the values in the set which are in 1:1 correspondence, I must get the exact value of the irrational number -- and not some value that is missing some of the digits. (This is a set, not a series; so were not taking a limit -- we're adding everything up in the set.)

So I think cantor means that kind of infinity, not finite unbounded -- but an actual infinite set capable of handling infinite digit values, and not just an approximation but in a countable ordering.

Irr

Irr

...

Therefore: If the set has an infinite cardinality, then the sum of all the elements must equal the value of the irrational number, not some subset of it -- or a finite but unbounded amount less than the irrational numbers value -- but rather, the true sum of everything in the set :: otherwise, the set's not infinite but truncated at some point short of all the digit values -- even if we couldn't determine exactly how far out it was truncated.

Do I have this right? Infinite means infinite, all, forever, nothing short of it... ? That's what cantor means?

If this is so, and Cantor means it really is infinite, then it seems I should also be able to use 1:1 correspondence to inductively prove a construction could contain a subset with cardinality aleph naught; (which isn't a big deal at first sight... -- but let's do it, so I can show the obvious -- and then after I prove this, I'll show where even though I was careful to do this exactly like the counting set, that 1:1 correspondence allows me to prove something that is unexpected and I'm not sure what it means. )

The task: Construct a set that by it's very nature, must by 1:1 correspondence, contain an element of cardinality -- ℵ. The set will be called "O

So, I'm going to do this the exact same way I did the construction of the set of counting numbers, but I'll count by the cardinality of the largest element in the previous set + 1, rather than by an integer number. It's a trivial change -- but it also will show how solidly the proof ties in with the very notion of counting by correspondence. (listing).

O

The next step of the construction is always to add '1' element to the element which will be unioned to the set.

O

O

O

and if allowed to continue forever, I have inductively constructed the set:

O

For, in any construction step it can be seen by inspection and definition that the cardinality of the element unioned has the same cardinality as the set created by that construction's step. Therefore the condition is ALWAYS met that for any step of the construction, XX - 1, that the next step of the construction XX will also fulfill the condition:

∃ N ∈ O

Therefore, inductively this condition must hold in the infinite case;

∃ N ∈ O

QED.

It's no big deal that an infinite set could contain another infinite set; and this didn't bother me at first -- but the problem is that the proof is based on 1:1 correspondence to determine that it contains infinite elements; and therefore it automatically paves a way to prove that any construction done in 1:1 correspondence (by counting) using a subset (any aleph naught subset) in place of Axx; if that subset has a supremium -- then O

eg: Consider the set 'irr' which I already described; and then Replace Axx in the above construction of O

What I'm getting at is that Either sets of cardinality Aleph naught must contain the supremium (if it exists) of any sequence used for counting by 1:1 correspondence, or else 1:1 correspondence does not prove a set length is truly infinite.

Which is really what Cantor means by infinity?

I'm not sure I understand.

For, it seems that Cantor sets of size Aleph naught must always be closed sets when it is possible they could be closed sets; but I get the impression that Cantor assumes the sets are open because his example set of counting numbers is an open set -- although I have never seen him state that his method only works on open sets, eg: ones that don't have a supremium.

Why is this?

For, it's not clear to me that 1:1 bijection can really prove something is infinite, rather than "finite unbounded" and I'd like someone to discuss the distinction, and how his tool of 1:1 correspondence really provides a valid and consistent tool that can be used in inductive proofs without fear of randomly confounding something that is really finite unbounded with something infinite.

eg: I'll show an inductive construction of the counting numbers, and then construct a proof based on the same principle that could potentially confound the two ideas (but doesn't have to) without making a mistake in the proof. (At least, I don't see a mistake...) and then could you explain to me exactly what kind of infinity Cantor had in mind for Aleph naught?

First off, consider what the set of counting numbers is:

The set of counting numbers, (A

_{ℵ}hereafter), is not something we can actually list all the elements of, physically; so the very idea of the set of all counting numbers is itself *inductively* constructed by making successive elements valued '1' larger than the previous element added to the set; eg: for every element, N, that is added, one adds the successor element who's value is N+1... literally, that's the idea of "counting" which gives the set it's name.eg: I would construct it like this for clarity.

N

_{a}=1 and A_{a}:= {} ⋃ { N_{a}} = { 1 }The next step of the construction is always to add '1' to the last counting number that was united to the set.

N

_{b}=N_{a}+1 and A_{b}:= A_{a}⋃ { N_{b}} = { 1 2 }N

_{c}=N_{b}+1 and A_{c}:= A_{b}⋃ { N_{c}} = { 1 2 3 }N

_{d}=N_{c}+1 and A_{d}:= A_{c}⋃ { N_{d}} = { 1 2 3 4 }N

_{e}=N_{d}+1 and A_{e}:= A_{d}⋃ { N_{e}} = { 1 2 3 4 5 }And if the iterative process is allowed to go on forever, the set constructed is by definition the set of all counting numbers who's cardinality is therefore ℵ

_{0}.A

_{ℵ}= { 1 2 3 4 5 6 7 8 ... }, |A_{ℵ}| = ℵ_{0}When Cantor claimed this set size/cardinality to be the definition of the first infinity, I think he meant to say that the set contains (eg: *inside* the set) an infinite number of elements by axiom; and not by proof. And I am, of course, therefore not asking him to prove it -- but I am concerned that this has unintentional side effects -- for infinity is NOT the same as "finite but unbounded" in traditional set theory, eg: the individual elements of the set of counting numbers are each finite -- but by the same token (finite elements) I simply couldn't prove that the set is infinite inductively, but Cantor gives me an axiom with which I supposedly can always accurately measure when something is infinite; and exactly so -- not plus or minus one unmatched element; and I want to explore what that really means -- for I'm unsure that I truly understand it, or him.

For I'm pretty sure Cantor is claiming that he is measuring an *infinite* number of elements, whenever a set has a inductively provable 1:1 correspondence with counting numbers. He doesn't intend a size that is finite, even if we don't know how large.

For example; if I had all the decimal digit values of an irrational number that happens to have no zero digit places, and contained in a sequence for convenience, but not strictly necessary -- (eg: a sequence, is a set which in this example I'm making the successors be in strictly decreasing magnitude), and this set contains a supremium if there is one ... "0" ? but even if it doesn't it wouldn't affect the value -- for all nonzero values are in the set.

irr = { 3 .1 .04 .001 .0005 ... }

Then clearly that set would be infinite in length, for the value is irrational and the digit places summing up to the irrational value, by definition, can never end; Thus, such a sequence by even being listable is clearly in 1:1 correspondence with counting numbers; and the cardinality is aleph naught.

So, if I add up ALL the values in the set which are in 1:1 correspondence, I must get the exact value of the irrational number -- and not some value that is missing some of the digits. (This is a set, not a series; so were not taking a limit -- we're adding everything up in the set.)

So I think cantor means that kind of infinity, not finite unbounded -- but an actual infinite set capable of handling infinite digit values, and not just an approximation but in a countable ordering.

Irr

_{a}= { 3}Irr

_{b}= { 3 .1 }...

Therefore: If the set has an infinite cardinality, then the sum of all the elements must equal the value of the irrational number, not some subset of it -- or a finite but unbounded amount less than the irrational numbers value -- but rather, the true sum of everything in the set :: otherwise, the set's not infinite but truncated at some point short of all the digit values -- even if we couldn't determine exactly how far out it was truncated.

Do I have this right? Infinite means infinite, all, forever, nothing short of it... ? That's what cantor means?

If this is so, and Cantor means it really is infinite, then it seems I should also be able to use 1:1 correspondence to inductively prove a construction could contain a subset with cardinality aleph naught; (which isn't a big deal at first sight... -- but let's do it, so I can show the obvious -- and then after I prove this, I'll show where even though I was careful to do this exactly like the counting set, that 1:1 correspondence allows me to prove something that is unexpected and I'm not sure what it means. )

The task: Construct a set that by it's very nature, must by 1:1 correspondence, contain an element of cardinality -- ℵ. The set will be called "O

_{ℵ}" for obvious... and I will prove that it has the property: ∃ N ∈ O_{ℵ}: |N| = |O_{ℵ}| = ℵ_{0}So, I'm going to do this the exact same way I did the construction of the set of counting numbers, but I'll count by the cardinality of the largest element in the previous set + 1, rather than by an integer number. It's a trivial change -- but it also will show how solidly the proof ties in with the very notion of counting by correspondence. (listing).

O

_{a}:= {} ⋃ { A_{a}} = { {1} }The next step of the construction is always to add '1' element to the element which will be unioned to the set.

O

_{b}:= O_{a}⋃ { A_{b}} = { {1} {1 2} }O

_{c}:= O_{b}⋃ { A_{c}} = { {1} {1 2} {1 2 3} }O

_{d}:= O_{c}⋃ { A_{d}} = { {1} {1 2} {1 2 3} {1 2 3 4} }and if allowed to continue forever, I have inductively constructed the set:

O

_{ℵ}= { A_{ℵ}{1} {1 2} {1 2 3} {1 2 3 4} ... }For, in any construction step it can be seen by inspection and definition that the cardinality of the element unioned has the same cardinality as the set created by that construction's step. Therefore the condition is ALWAYS met that for any step of the construction, XX - 1, that the next step of the construction XX will also fulfill the condition:

∃ N ∈ O

_{XX}: |N| = |O_{XX}|Therefore, inductively this condition must hold in the infinite case;

∃ N ∈ O

_{ℵ}: |N| = |O_{ℵ}| = ℵQED.

It's no big deal that an infinite set could contain another infinite set; and this didn't bother me at first -- but the problem is that the proof is based on 1:1 correspondence to determine that it contains infinite elements; and therefore it automatically paves a way to prove that any construction done in 1:1 correspondence (by counting) using a subset (any aleph naught subset) in place of Axx; if that subset has a supremium -- then O

_{ℵ}contains the supremium of the set.eg: Consider the set 'irr' which I already described; and then Replace Axx in the above construction of O

_{ℵ}, with irr_{xx}instead; the very proven fact that the set O_{ℵ}must contain an element with cardinality of aleph naught would automatically prove that O_{ℵ}must contain the set irr itself, and NOT just part of the set, and not missing even one element;What I'm getting at is that Either sets of cardinality Aleph naught must contain the supremium (if it exists) of any sequence used for counting by 1:1 correspondence, or else 1:1 correspondence does not prove a set length is truly infinite.

Which is really what Cantor means by infinity?

I'm not sure I understand.

For, it seems that Cantor sets of size Aleph naught must always be closed sets when it is possible they could be closed sets; but I get the impression that Cantor assumes the sets are open because his example set of counting numbers is an open set -- although I have never seen him state that his method only works on open sets, eg: ones that don't have a supremium.

Why is this?

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