Cont: Deeper than primes - Continuation 2

[doronshadmi]

(Curious, though.

Curiosity is clearly not one of your features , in this case.

 

[doronshadmi]

The valuation of 0.999... stands at 1.

If observed only from |N|.

http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110 is simply not in your |N|_only scope.

jsfisher, you simply have no argument anymore, so you are digging in the past and desperately try to convince yourself that real mathematics fits to your |N|_only scope.

Doron, you do not get to redefine the meanings

I do not redefine anything. My argument is true if |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... is true.

Simple as that, as clearly shown in http://www.internationalskeptics.com/forums/showpost.php?p=10343813&postcount=202.

Your ∞ hands waving does not hold water.

 

[jsfisher]

If observed only from |N|.

Doesn't matter how it is "observed." Doron, you don't get to redefine mathematics to accommodate your personal confusion.

 

[doronshadmi]

Doesn't matter how it is "observed." Doron, you don't get to redefine mathematics to accommodate your personal confusion.

Dear jsfisher.

This is the beautiful thing here, I do not redefine anything.

 

[jsfisher]

Dear jsfisher.

This is the beautiful thing here, I do not redefine anything.

Sure you did. The decimal notation 0.999... has a well-defined meaning, that of a series. The series has a value determined by the limit of the partial summation sequence corresponding to the series. The determination of limits is well-defined.

The consequence of all that is that 0.999... is identical in value to 1.

Your attempt to "observe" it differently is a redefinition. I can observe 4 differently as 5; that doesn't make it 5. Moreover, your appeal to cardinal numbers is unwarranted since they have no function in establishing the value of 0.999....

 

[realpaladin]

Dear jsfisher.

This is the beautiful thing here, I do not redefine anything.

If all the colors are only to be seen through a new blue lens, then all the colors are redefined as shades of blue.

 

[doronshadmi]

Sure you did.

Sure I did not.

Your attempt to "observe" it differently is a redefinition.

Not at all, it is based on the well defined mathematical fact that |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ...

I can observe 4 differently as 5; that doesn't make it 5.

Even in this simple case you fail. Observing 4 from 5 simply enables one to know that 5 > 4. There is nothing in this observation that can be interpreted as if 4 is 5.

Moreover, your appeal to cardinal numbers is unwarranted since they have no function in establishing the value of 0.999....

They have a function in establishing the value of 0.999..., but it clearly not in the scope of one that interpretes 4 as 5.

 

[jsfisher]

Not at all, it is based on the well defined mathematical fact that |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ...

You keep repeating that, but you never actually show how it (whatever "it" actually is) is based on those inequalities. How do you connect Cantor's Theorem to the definition for limits in such a way that 0.999... and 1 are not identical?

 

[doronshadmi]

You keep repeating that, but you never actually show how it (whatever "it" actually is) is based on those inequalities.

All you have to do is to deduce a given mathematical framework in terms of cardinality, in order to provide a given solution.

For example:

Finite or infinite cardinality:

A given solution that is satisfied by at least some cardinality > |1|, is not satisfied by some cardinality ≤ |1|.

Infinite cardinality:

A given solution that is satisfied by at least some cardinality > |N|, is not satisfied by some cardinality ≤ |N|.

A given solution that is satisfied by at least some cardinality > |P(N)|, is not satisfied by some cardinality ≤ |P(N)|.

A given solution that is satisfied by at least some cardinality > |P(P(N))|, is not satisfied by some cardinality ≤ |P(P(N))|.

A given solution that is satisfied by at least some cardinality > |P(P(P(N)))|, is not satisfied by some cardinality ≤ |P(P(P(N)))|.

etc. ... at infinitum.

How do you connect Cantor's Theorem to the definition for limits in such a way that 0.999... and 1 are not identical?

A given solution that is satisfied by at least some cardinality > |N|, is not satisfied by some cardinality ≤ |N|.

 

[jsfisher]

You keep repeating that, but you never actually show how it (whatever "it" actually is) is based on those inequalities.

All you have to do is to deduce a given mathematical framework in terms of cardinality, in order to provide a given solution....

So you are unable to make clear whatever "it" actually is.

Telling us all what it's based on and could be deduced from is just hand-waving without ever defining "it".

How do you connect Cantor's Theorem to the definition for limits in such a way that 0.999... and 1 are not identical?

A given solution that is satisfied by at least some cardinality > |N|, is not satisfied by some cardinality ≤ |N|.

And this is relevant to the definition for limits just how?

 

[realpaladin]

A given solution that is satisfied by at least some cardinality > |N|, is not satisfied by some cardinality ≤ |N|.

I am not entirely sure why the at least is emphasized by an underscore...

The greater than symbol already signifies this.

And as for the logic... this is the same as stating: If it is not black, then it is a different color...

 

[jsfisher]

I am not entirely sure why the at least is emphasized by an underscore...

The greater than symbol already signifies this.

And as for the logic... this is the same as stating: If it is not black, then it is a different color...

And you aren't even going to ask what it means to satisfy a solution, now are you? Nor how a solution might be satisfied by a cardinal number, either, right?

 

[doronshadmi]

So you are unable to make clear whatever "it" actually is.

Telling us all what it's based on and could be deduced from is just hand-waving without ever defining "it".

This "it" is simply cardinality, and it measures the minimal needed values that satisfy a given value, for example:

0.9+0.09+0.009 that has |3| values can't satisfy value 1, and in this case at least 0.9+0.09+0.009+... that has |N|+1 = |N| values, satisfies value 1.

In case that the minimal needed values that satisfy 1 is at least |P(N)|+1 = |P(N)|, 0.9+0.09+0.009+... that has |N|+1 = |N| values, can't satisfy value 1.

In case that the minimal needed values that satisfy 1 is at least |P(P(N))|+1 = |P(P(N))|, no |P(N)|+1 = |P(N)| of such values satisfy value 1.

Etc. ... ad infinitum.

And this is relevant to the definition for limits just how?

It is about the ability to satisfy a given value, as explained above.

 

[jsfisher]

This "it" is simply cardinality, and it measures the minimal needed values that satisfy a given value

Just how do values satisfy a given value? What values would satisfy 42, just as an example?

for example:

0.9+0.09+0.009 that has |3| values

No, that is an expression of three values (and, no, I don't need to extract the absolute value of 3). The expression has a value, and that is 0.999.

...can't satisfy value 1

"Satisfy value 1"? Perhaps you meant "equal" in place of "satisfy value"?

...and in this case at least 0.9+0.09+0.009+... that has |N|+1 = |N| values, satisfies value 1.

0.9 + 0.09 + 0.009 + ... is an infinite series. The only thing important in determining the value of the series is the limit of the related sequence of partial summations. No infinity is needed to establish that the limit is 1 and thus 0.999... is identical to 1.

(By the way, your assertion of |N| + 1 terms are involved is just bizarre. So is your usage of the phrase, "at least".)

In case that the minimal needed values that satisfy 1 is at least |P(N)|+1 = |P(N)|, 0.9+0.09+0.009+... that has |N|+1 = |N| values, can't satisfy value 1.

And what case would that be? Again, series, limits, value without every visiting any flavor of infinity.

...
It is about the ability to satisfy a given value, as explained above.

Your use of that highlighted word is novel.

 

[doronshadmi]

(By the way, your assertion of |N| + 1 terms are involved is just bizarre.

The bizarre thing here is actually your mathematical framework that excludes the different values of cardinality as an essential factor of a given solution.

As a result http://www.internationalskeptics.com/forums/showpost.php?p=10318337&postcount=7, http://www.internationalskeptics.com/forums/showpost.php?p=10318659&postcount=9, http://www.internationalskeptics.com/forums/showpost.php?p=10332081&postcount=110 and http://www.internationalskeptics.com/forums/showpost.php?p=10353611&postcount=233 are not in the scope of your mathematical framework.

Moreover, the treatment of your framework about infinity (as seen in http://www.internationalskeptics.com/forums/showpost.php?p=10344228&postcount=203) simply excludes the different values of transfinite cardinality as an essential factor of a given solution.

Since your framework excludes the different values of cardinality as an essential factor of a given solution, there can't be any communication between us about this fine subject.

You can enjoy your ∞ hand-waving as much as you like, which uses each cardinal number separately from the other cardinal numbers, but you will not find my framework under your hand-waving.

Here is a concrete example of your hand-waving, by using your own words

0.9 + 0.09 + 0.009 + ... is an infinite series. The only thing important in determining the value of the series is the limit of the related sequence of partial summations. No infinity is needed to establish that the limit is 1 and thus 0.999... is identical to 1.

that simply excludes the different values of cardinality (and in this case, transfinite cardinality) as an essential factor of a given solution.

 

[realpaladin]

And you aren't even going to ask what it means to satisfy a solution, now are you? Nor how a solution might be satisfied by a cardinal number, either, right?

It's Doron... I am waaay beyond that already...

 

[doronshadmi]

http://www.internationalskeptics.com/forums/showpost.php?p=10316119&postcount=4298 is defined by using different values of cardinality as an essential factor of a given solution, and it is definitely not in the scope of a framework that uses only each cardinal number separately from the other cardinal numbers, in order to provide a given solution.

0.9 + 0.09 + 0.009 + ... is an infinite series.

More accurately, 0.9 + 0.09 + 0.009 + ... is an infinite series of countable |N| values and it is < 1 if it used among an infinite series of uncountable |P(N)| values.

In order to understand it all you have to do is to realize, for example, how 0.9 + 0.09 + 0.009 finite series of countable |3| values is < 1 if it used among an infinite series of countable |N| values like 0.9 + 0.09 + 0.009 + ...

The same principle holds for both cases.

 

[psionl0]

0.999... stands at 1.

Just to be technically accurate, the limit of 0.999... stands at 1. No matter how many 9's you add to the string, the sum will never add up to 1.

0.999... is a convergent countably infinite series. Trying to make more of it than that like doronshadmi is doing or making an unprovable assertion that an infinite string of 9's would still add up to < 1 is to just generate meaningless confusing jargon.

 

[realpaladin]

You can enjoy your ∞ hand-waving as much as you like, which uses each cardinal number separately from the other cardinal numbers, but you will not find my framework under your hand-waving.

Some true words by Doron... I thoroughly enjoy the rigorously proven thing that he denotes by 'hand-waving', and I sure have not found even the cornerstone of a framework by Doron in any of the 10k+ posts in over 7 years.

Doron, again, it is not the fault of the students (there have been so many, with so many different views; from Apathia who wishes you the best but does not understand your words, via Dessi who is your superior in mathematics but still can't follow you, to JsFisher and myself who are just tenacious in trying you to commit to something that is defined), but the fault of the teacher if the knowledge is never passed on.

Because even *if* we all were to agree you are right, you just wait until there is something to kibitz about; you think that discussion and kibitzing makes you look smart... it does not, it makes you look quarrelsome and angry.

 

[psionl0]

My terms are rigorously defined by the fact that |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ...

As your diagram in http://www.internationalskeptics.com/forums/showpost.php?p=10334974&postcount=158 showed, this is true only as long as N is some finite subset of the set of natural numbers. You haven't demonstrated this to be true for a countably infinite set of natural numbers.

I'd go as far as to say that if N is the set of all natural numbers, then all of the supersets that you generate are also countably infinite.