Cont: Deeper than primes - Continuation 2

[Little 10 Toes]

Define "Archimedean property". Use non-doron definitions.

 

[doronshadmi]

Define "Archimedean property". Use non-doron definitions.

I am using the standard definition of Archimedean property (https://en.wikipedia.org/wiki/Archimedean_property) in this case, and my number system does not satisfy it.

 

[Nonpareil]

If one takes the Archimedean property as a religious dogma, then what is written above is inevitable.

The Archimedean property has nothing to do with the fact that you don't understand what the phrase "countably infinite" means.

Try again.

 

[jsfisher]

The Archimedean property has nothing to do with the fact that you don't understand what the phrase "countably infinite" means.

Try again.

He doesn't understand his latest Google discovery either.

 

[MetalPig]

Amongst all the wrong, this one may be correctable: radix.

I have redixed my potatoes. Somehow they became undixed overnight.

 

[doronshadmi]

The Archimedean property has nothing to do with the fact that you don't understand what the phrase "countably infinite" means.

A set is "countably infinite" if there is a bijection between its members and the members of the set of natural numbers.

By the Archimedean property there are no infinitely large or infinitely small elements, (where in the considered case an element is understood as number).

If one takes the Archimedean property as a religious dogma, one simply can't deal with infinitely large numbers as a replacement of transfinite cardinal numbers like aleph₀, which is the cardinality of the set of natural numbers, according to the Archimedean property.

So, as you see Nonpareil, there is a lot to do with the phrase "countably infinite" and the Archimedean property.

 

[doronshadmi]

All of them have a 1 and infinitely many 0s left of the radix point, so there is no meaningful way of telling if my 1,000,... is bigger or smaller than your 1,000,...
Putting a space in front of the number doesn't usually change the value. If it does in your numbers system, your system is silly.

MetalPig, it is known (by using the standard notions about, what is known as transfinite numbers) that one can't distinguish between, for example, aleph₀+1 and aleph₀.

Does aleph₀ is not a silly number exactly because one can't distinguish between aleph₀+1 and aleph₀?

If your answer is yes, then please explain why aleph₀+1 must be indistinguishable of aleph₀ in order to understand them in terms of a non-silly number system?

 

[jsfisher]

...
By the Archimedean property there are no infinitely large or infinitely small elements, (where in the considered case an element is understood as number).
...

You might want to look a bit closer at what this actually means before you declare your (still ill-defined) number system to be non-Archimedean.

 

[doronshadmi]

You might want to look a bit closer at what this actually means before you declare your (still ill-defined) number system to be non-Archimedean.

1/n > 1/1,000,000,000,... is exactly non-Archimedean.

Once again, here is some concrete example:

1,000,000,000,... can be considered as a number that is greater than any natural number, such that number h of non-standard analysis has the value 1/1,000,000,000,... < any value of the form 1/n

Any attempt to deal with such numbers in terms of Archimedean proprty, is indeed ill-defined.

 

[doronshadmi]

Moreover jsfisher, let's not forget the following:

It has to be stressed that (There is no infinite position) AND (There are infinitely many such positions) - as said in http://www.internationalskeptics.com/forums/showpost.php?p=11246979&postcount=1080 - is a logical contradiction, simply because we are talking about the same object that has infinitely many positions at the right side of the radix point (as seen, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=11246733&postcount=1075).

 

[jsfisher]

Moreover jsfisher, let's not forget the following...

You needn't worry. No one has forgotten how wrong you were.

 

[Nonpareil]

By the Archimedean property there are no infinitely large or infinitely small elements, (where in the considered case an element is understood as number).

The Archimedean property is a property of certain specific fields, doron. Since we are specifically talking about infinitely large or small quantities, it does not apply here - which is why no one brought it up. No one is making any arguments based on assuming that it applies to the systems in play here. No one cares - except you, apparently, since you don't actually understand the arguments being put against you and are fond of simply throwing impressive-sounding terms around without understanding what they mean.

Unfortunately, the people you are speaking to do understand those terms, and are not impressed.

If one takes the Archimedean property as a religious dogma, one simply can't deal with infinitely large numbers as a replacement of transfinite cardinal numbers like aleph₀, which is the cardinality of the set of natural numbers, according to the Archimedean property.

The incoherence of your attempts to replace Aleph-0 with a non-equivalent, insufficiently-defined, poorly-notated string of digits that explicitly do not behave equivalently to Aleph-0 has nothing to do with the Archimedean property.

If your answer is yes, then please explain why aleph₀+1 must be indistinguishable of aleph₀ in order to understand them in terms of a non-silly number system?

Because Aleph-0 is a measure of the size of an infinite set. Add one to infinity and you still have infinity.

Please do some basic reading on concepts like this before trying to make them central to your arguments. It will save us all a lot of time and headaches.

And, to make sure it is not missed, the relevant snippet from the above link:

"So, we've got aleph-null, the set of all natural numbers. Now, which is bigger: aleph-null, or aleph-null+1? The old "just add 1" canard comes up all the time when we're talking about the largest finite numbers, and with good reason - you can always just add 1 to a finite number and come up with something even bigger. But does that work for aleph-null? Well, let's borrow the tuna sandwich from our earlier set and add it to the set of all natural numbers, so we've now got a set with aleph-null+1 terms.

As we've established, the only way to compare these two sets is with one-to-one correspondence. We'll put the tuna sandwich at the start of one set, which we'll call Set C, while Set D will just be the standard set of natural numbers. So then, Set C begins {tuna sandwich, 0, 1, 2, 3, 4...}, while Set D is {0, 1, 2, 3, 4, 5...}. We'll match the tuna sandwich to 0, 0 to 1, 1 to 2, 2 to 3, 3 to 4, 4 to 5...and so on forever. After all, there are still infinitely many terms in both sets, and we can keep up the one-to-one correspondence for as long as we like without ever running out of terms. That means aleph-null and aleph-null plus a tuna sandwich are precisely equal."

Moreover jsfisher, let's not forget the following:

It has to be stressed that (There is no infinite position) AND (There are infinitely many such positions) - as said in http://www.internationalskeptics.com/forums/showpost.php?p=11246979&postcount=1080 - is a logical contradiction

Repeating your failure to understand the concept of countable infinity is not going to get you anywhere. It was wrong the first time, and it continues to be wrong no matter how many times you reiterate it.

 

[doronshadmi]

You needn't worry. No one has forgotten how wrong you were.

Once again no detailed reply about the last discussion.

Your evasion has been noted.

 

[doronshadmi]

"let's borrow the tuna sandwich from our earlier set and add it to the set of all natural numbers, so we've now got a set with aleph-null+1 terms."

No dear Nonpareil, the tuna sandwich is beyond the range of (what you call) the set of all natural numbers, exactly as s that is mapped with 10111010011... is beyond the range of all the s members in the following denationalization:

So, there is no such thing like a complete set of infinitely many elements, or in other words, an infinitely large number like
1,000,000,000,... < 1+1,000,000,000,... by 1, where an infinitely large number like 1,000,000,000,... does not satisfy the Archimedean property.

In other words, if s is added to the set of all natural numbers, it means that what is considered as the set of all natural numbers is actually not the set of all natural numbers, in the first place, so there is no such thing like bijection with the set of all natural numbers, and the notion of "countably infinite" set actually does not hold.

So http://gizmodo.com/5809689/a-brief-introduction-to-infinity is based on the wrong notion of a complete infinite set.

Moreover, Alasdair Wilkins's posts on Math are poorly written, even by using the standard (and wrong) notion of a complete infinite set (see the discussion at the end of http://gizmodo.com/5809689/a-brief-introduction-to-infinity), so please carefully look what stuff you are going to use, before you are using it.

 

[Nonpareil]

No dear Nonpareil, the tuna sandwich is beyond the range of the set of all natural numbers

Thank you for demonstrating so readily that you neither understood nor care to understand the information given to you.

What you add to an infinite set does not matter. So long as the addition is finite, it does not alter the size of the set.

 

[jsfisher]

...
if s is added to the set of all natural numbers
...

Why would one do that? S isn't a natural number, so if one were to add it to the set of natural numbers, what utility would the resultant set have?

 

[doronshadmi]

Thank you for demonstrating so readily that you neither understood nor care to understand the information given to you.

What you add to an infinite set does not matter. So long as the addition is finite, it does not alter the size of the set.

There is not such thing like a complete infinite set, in the first place, but you don't care about this fact.

 

[jsfisher]

There is not such thing like a complete infinite set, in the first place, but you don't care about this fact.

Why didn't you just say you reject the Axiom of Infinity in the first place. Makes for a dull set theory, though.

 

[doronshadmi]

Why would one do that? S isn't a natural number

s isn't a natural number according to the notion that there is a complete set of natural numbers, exactly because such notion is based on the Archimedean property, which according to it there is no infinitely large number like 1,000,000,000,...

Cantor's diagonal proof by contradiction can't be used under non-Archimedean number system.

 

[doronshadmi]

Why didn't you just say you reject the Axiom of Infinity in the first place. Makes for a dull set theory, though.

I do not reject the existence of infinitely large numbers, as you do.

As a result the size of infinitely many elements like natural numbers does not satisfy the Archimedean property (there can be infinitely many incomplete sets of natural numbers, where each set has a different infinite size)