Cont: Deeper than primes - Continuation 2

[doronshadmi]

They just keep going.

0...1 infinite logical connective indeed has infinitely many 0 bits that just keep going.

You simply avoid http://www.internationalskeptics.com/forums/showpost.php?p=11484090&postcount=2119.

 

[jsfisher]

Doronshadmi,

No, there are not. Infinite sequences do not come to a abrupt end somewhere out there. They just keep going. Otherwise, they wouldn't be infinite.

You are dealing with fantasy, not Mathematics. Let me know if and when you want to discuss the latter.

 

[doronshadmi]

jsfisher,

You simply avoid http://www.internationalskeptics.com/forums/showpost.php?p=11483843&postcount=2115, http://www.internationalskeptics.com/forums/showpost.php?p=11483887&postcount=2117, http://www.internationalskeptics.com/forums/showpost.php?p=11484090&postcount=2119 and http://www.internationalskeptics.com/forums/showpost.php?p=11484108&postcount=2121 exactly because they do not fit to the standard notions since the mathematicians that developed the standard notions, did not study infinite logical trees.

Let me know if you are open to study new mathematical/logical notions (actually keep going and not just talking about keep going).

 

[jsfisher]

...exactly because they do not fit to the standard notions since the mathematicians that developed the standard notions, did not study infinite logical trees.
Let me know if you open to study new mathematical/logical notions.

Really? Gee, I had always thought they were (and continue to be) studied extensively. There are even names for various forms of infinite trees.

There is a lot published about them. Who knew they hadn't been studied at all?

 

[doronshadmi]

Really?

Really.

Please provide a link the some work that extensively studies infinite logical connectives.

 

[doronshadmi]

Pi in base 2 starts by 11.001001000011111101101010100010001000010110100011000010001101001100010...

Do we need to write down all the 0;1 bits in order to define this infinite string as pi by traditional mathematics?

The answer by traditional mathematicians is generally: "No, we don't have to do that since the abstraction of mathematics enables to define mathematical objects even if they are not explicitly constructed".

--------

Now I am going to do something that will immediately be rejected by most if not all traditional mathematicians, which is to write down an ordered list of infinite logical connectives, such that each infinite logical connective has its, so called, immediate successor.

First let's define ... as a notation of infinitely many existing but non-written bits, or also as a notation of infinitely many existing but non-written infinite logical connectives.

The rest can be found here

Edited by kmorts: 

Removed to comply with Rule 4

 

[MetalPig]

Oh good, the infinitely long numbers with two ends are back. Or maybe they never went away; I can't tell.

 

[doronshadmi]

Oh good, the infinitely long numbers with two ends are back. Or maybe they never went away; I can't tell.

First of all they are infinitely long ordered logical connectives from contradiction to tautology that are unbounded (or as jsfisher says: "just keep going") from within.

Take for example the following set's representation: {{}, {1}, ..., {2}}. It does not prevent the fact that it can be an infinite set.

Moreover, take the interval [0,1]. It allays a room for infinite amount of ordered distinct points.

 

[doronshadmi]

Generally, the problem to understand my notions of infinite cardinality arises if one defines it in therms of a fixed size like aleph₀.

But as very simply shown in http://www.internationalskeptics.com/forums/showpost.php?p=11480438&postcount=2098 infinite cardinality is not a fixed size, but it is a spectrum of infinitely many proportions that are kept w.r.t each other during the "keep going" inherent property of any given infinite set.

The transitions between proportions is actually the transitions between distinct logical paths of a given infinite ordered logical tree (or a list, as given in http://www.internationalskeptics.com/forums/showpost.php?p=11484623&postcount=2126).

 

[jsfisher]

Moreover, take the interval [0,1]. It allays a room for infinite amount of ordered distinct points.

That word does not mean what you think it means.

You cannot, for example, list the members of [0,1] in order, nor can you identify a number that immediately precedes (or follows) any other number within the interval.

And no amount of magic derived from the strategic placement of ellipses suddenly gives infinite sequences final elements.

 

[doronshadmi]

That word does not mean what you think it means.

You cannot, for example, list the members of [0,1] in order, nor can you identify a number that immediately precedes (or follows) any other number within the interval.

And no amount of magic derived from the strategic placement of ellipses suddenly gives infinite sequences final elements.

This is a constructivist view.

By non-constructivist view there is a spectrum of logical connectives from 0 (contradiction) to 1 (tautology) similar to Fuzzy logic.

Because of your constructivist view you are missing also http://www.internationalskeptics.com/forums/showpost.php?p=11486500&postcount=2129.

Indeed the word order by non-constructivist view does not mean what you think it means by your constructivist view.

 

[jsfisher]

Indeed the word order by non-constructivist view does not mean what you think it means by your constructivist view.

You have tried this silliness before, and it didn't work then, either. You don't get to change the meaning of well-established terms for your own misguided purposes.

The points in [0,1] are well-ordered; you cannot list them in order.

If there is a different concept you'd like to use, feel free to define it; just be so kind as to give it a new name, too. And, by the way, no matter what you name your term and how you define it, you will still not be able to list the points in [0,1].

 

[doronshadmi]

You have tried this silliness before, and it didn't work then, either. You don't get to change the meaning of well-established terms for your own misguided purposes.

The points in [0,1] are well-ordered; you cannot list them in order.

If there is a different concept you'd like to use, feel free to define it; just be so kind as to give it a new name, too. And, by the way, no matter what you name your term and how you define it, you will still not be able to list the points in [0,1].

The following infinite logical tree is the spectrum (and therefore infinitely many ordered logical connectives) from contradiction to tautology:

*
|\
| \
|  \
|   \
|    \
|     \
|      \
|       \
|        \
|         \
|          \
|           \
|            \
|             \
|              \
0               1 
|\              |\
| \             | \
|  \            |  \
|   \           |   \
|    \          |    \
|     \         |     \         
|      \        |      \
0       1       0       1
|\      |\      |\      |\
| \     | \     | \     | \
|  \    |  \    |  \    |  \
0   1   0   1   0   1   0   1
|\  |\  |\  |\  |\  |\  |\  |\
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

              ...

By using a non-constructivist approach one immediately notices that this spectrum can be also arranged as follows:

Let _._._._ be a notation for infinitely many 0;1 bits or infinitely many distinct paths that are not explicitly written but (in your words jsfisher "just keep going") from within.

In that case the infinite logical spectrum of the tree above is listed as follows:

0_._._._0
0_._._._1
0_._._._0
0_._._._1
0_._._._0
0_._._._1

_._._._

1_._._._0
1_._._._1
1_._._._0
1_._._._1
1_._._._0
1_._._._1

and there is no problem to index each one of them by a given N member.

In other words, the cardinality of N is exactly the spectrum from contradiction to tautology.

So jsfisher, you can use your established "well-ordered" as much as you like, but it is not the spectrum (and therefore infinitely many ordered logical connectives) from contradiction to tautology.

 

[jsfisher]

...nonsense snipped...

Retreat to any special point of view you like, but infinite sequences still won't have a final element.

 

[doronshadmi]

Retreat to any special point of view you like, but infinite sequences still won't have a final element.

Since the cardinality of N is exactly the spectrum from contradiction (0...0) to tautology (1...1), each infinite logical connective has first AND last bits, but the number of bits is infinite, similar to Fazzy logics, which defines the degree of membership by R members from 0 (fully not a member) to 1 (fully a member).

Actually the notion of infinite fixed cardinality like aleph₀ for set N, naturally prevents the notion of infinite spectrum of infinite cardinalities for set N.

...nonsense snipped...

A typical response of one that has a constructivist view of the issue at hand.

-------------

Generally in your mathematical universe cardinality is not defined by spectrum of infinite logical connectives from contradiction (0...0) to tautology (1...1) so you can't value Cardinality as defined in my mathematical universe, by using notions that are taken from your mathematical universe. Specially you can't use the notion of aleph₀ in my mathematical universe.

 

[jsfisher]

Since the cardinality of N...

Repeating an incorrect statement doesn't change its truth value. Infinite sequences still have not final element.

 

[doronshadmi]

Repeating an incorrect statement doesn't change its truth value. Infinite sequences still have not final element.

In my mathematical universe last is not the same as final.

Repeating a statement that is done in your mathematical universe doesn't change the truth values (the infinite spectrum of logical connectives as the logical basis of Cardinality) in my mathematical universe.

 

[abaddon]

In my mathematical universe last is not the same as final.

Who defines what the last element of any infinite set is? You?

 

[jsfisher]

Who defines what the last element of any infinite set is? You?

He's desperately trying to hide the infinity in the middle so he doesn't have to deal with it. Those ellipses are magic.

Were he to structure it with two ends and an infinity between (which his binary tree model does not allow), he'd still be left with too deep a well to climb back out of to build his list.

No matter which way Doronshadmi lies to himself, he cannot list an uncountable set.

 

[doronshadmi]

Who defines what the last element of any infinite set is? You?

Do you see the interval [0,1] ?

It has first AND last elements, but still it has infinitely many elements in between which their cardinalities are a spectrum that is derived from infinitely many distinct logical connectives that each one of them has first AND last bit exactly as the interval [0,1] has first AND last elements.

The whole structure is logically consistent.