All the possible posts in PF, all the possible human thoughts

[Delta2]

Homework Statement

How many are all the possible posts that can be posted in a forum like PF, including posts that make no sense like for example "aka luga muga aghdlgadlkg.LSJDJSKalsdasldl 1-+=01./+"

Relevant Equations

order of n things per m=$n^m$, $|\bigcup A_i |=\sum |A_i|$

Well since we count all possible posts , including those that have no syntax, no proper grammar or proper words, and no sensible meaning, all posts of length n are something like 100^n (I suppose we have 100 possible chars besides the letters of ordinary alphabet, including symbols like the integral, = ,+- , space bar e.t.c). The total number of posts is $$\sum_{n=0}^{\infty} 100^n$$ which seems infinite but countable infinity.

Now if we assume that all human thoughts can be expressed by words, then the set of all human thoughts is a subset of the above set, hence they are countable infinite at most too.

What do you think? This is not a real homework btw, I am a retired high school teacher.

 

[FactChecker]

It is not hard to prove that the set of all possible [EDIT: finite] texts made from the character set is countably infinite. It seems harder to accept your assumption that all human thoughts can be expressed with complete accuracy by words.

 

[nuuskur]

If the alphabet was countable, then the set of all finite words is still countable. I suppose it is reasonable to expect that every thought can be expressed in finitely many symbols.

 

[Delta2]

Hm, yes one point is that whether or not human thoughts can be expressed totally with words or we still miss something no matter how many words we use.

And something else now that I went to bed to rest a bit and was thinking of this, the thought occurred my mind that this set might lead to some sort of paradox, as the set of all sets paradox. I mean if the set of all human thoughts is also a human thought (with infinite number of words or perhaps uncountable number of words, not sure).

 

[Ibix]

This is a book - The Library of Babel by Jorge Luis Borges. All his texts have a fixed character count, including spaces so shorter texts are possible by having many spaces at the end. Longer texts are possible by treating fixed length texts as volumes 1, 2, 3, etc. So all possible books are there.

The library consists of tesselated hexagonal cells with fixed numbers of books on fixed numbers of shelves in each cell. He calculates the number of cells. As far as I recall the narrator notes that no one has ever seen a comprehensible text (they are too rare) nor have they seen an edge to the library. Furthermore, he notes that the library is useless because there exists a text that says "it is a good idea to eat" and another which says "it is not a good idea to eat".

 

[FactChecker]

What about the idea of an irrational number like $\pi$? Some thoughts can be represented by a new symbol. Does that make the number of possible human thoughts uncountable?

 

[Delta2]

What about the idea of an irrational number like $\pi$? Some thoughts can be represented by a new symbol. Does that make the number of possible human thoughts uncountable?

$\pi$ can be represented by an infinite number of digits , so I think it is included as a possible post.

But hm, if $\pi$ is included then so is $e$ and so is every irrational number. But the set of all irrational numbers is uncountable.. Where did I go wrong?!??Hmm, I thought countable union of countable sets is a countable set but I guess this is where I went wrong.

 

[FactChecker]

$\pi$ can be represented by an infinite number of digits , so I think it is included as a possible post.

But hm, if $\pi$ is included then so is $e$ and so is every irrational number. But the set of all irrational numbers is uncountable.. Where did I go wrong?!??Hmm, I thought countable union of countable sets is a countable set but I guess this is where I went wrong.

Cantor's diagonalization argument shows that the set of all character strings including infinitely long strings is an uncountable set. So your original scenario must exclude infinitely long text strings. I think that is a reasonable assumption, but I am not sure that all possible thoughts can be included that way.

 

[pbuk]

For obvious reasons no forum can include infinitely long posts. The first limit that is likely to be hit is the maximum POST request size set in the web server's configuration. By default (Apache) this is 2MB, although the forum software can choose to set a lower limit on message length (and probably does for various reasons).

 

[Delta2]

For obvious reasons no forum can include infinitely long posts. The first limit that is likely to be hit is the maximum POST request size set in the web server's configuration. By default (Apache) this is 2MB, although the forum software can choose to set a lower limit on message length (and probably does for various reasons).

Ah yes I assumed that there is no limitation on the length of the posts. Now days hard disks and SSDs are cheap e hehe..

 

[pbuk]

Now days hard disks and SSDs are cheap e hehe..

But content that is too long for anyone to read is worthless and a waste of (relatively costly) bandwidth and indexing time.

 

[Hall]

I think this post (not mine, the original post) is simply a beautiful manifestation of evolved human mind.

 

[jbriggs444]

What about the idea of an irrational number like $\pi$? Some thoughts can be represented by a new symbol. Does that make the number of possible human thoughts uncountable?

$\pi$ can be represented by an infinite number of digits , so I think it is included as a possible post.

But hm, if $\pi$ is included then so is $e$ and so is every irrational number. But the set of all irrational numbers is uncountable.. Where did I go wrong?!??Hmm, I thought countable union of countable sets is a countable set but I guess this is where I went wrong.

$\pi$ is included because it is a finite string. One character if you allow the greek character $\pi$. Two characters if you use the two letter spelling "pi". Various finite algorithms that yield $\pi$ are valid as long as they can be presented in a finite post. The infinite string beginning with 3.14159... is not included because it is an infinite string and only finite posts are legal.

$e$ is included for similar reasons.

The set of all finite strings of all possible lengths, however, does not include any infinite strings. The countable union (over the set of all possible finite string lengths) of countable sets (the sets of strings of a particular length) is indeed countable. It has the cardinality of the set of natural numbers. But it does not include the complete decimal expansions of $\pi$, $e$ or of any other irrational number.

Cantor's proof demonstrates, among other things, that the set of all irrational numbers is uncountable. Since the set of finite representations is countable it follows that: For any given language, some irrational numbers (indeed "most" of them) will have no finite representation in that language.

 

[jbriggs444]

Ah yes I assumed that there is no limitation on the length of the posts. Now days hard disks and SSDs are cheap e hehe..

The gap between insanely large and unboundedly large is considerable. As is the gap between unboundedly large and infinite.

 

[Delta2]

The set of all finite strings of all possible lengths, however, does not include any infinite strings.

This statement is confusing to me. How can we have all possible lengths, 1,2,...,n,... but not infinite strings?

For any given language, some irrational numbers (indeed "most" of them) will have no finite representation in that language.

Tell me a language where $\pi$ or another irrational number has a finite representation.

 

[jbriggs444]

#This statement is confusing to me. How can we have all possible lengths, 1,2,...,n,... but not infinite strings?

Every natural number is finite. There are no infinite integers. (At least not in the standard model.

[Edit 2: There is more complexity in the idea of what constitutes a "model" than can reasonably be pursued here.

The Peano axioms are one starting point to understand the natural numbers.

Tell me a language where $\pi$ or another irrational number has a finite representation.

You just expressed $\pi$ with a finite representation: ##\pi##. One can also express $\sqrt{2}$ in that language: as ##\sqrt{2}##.

Edit: Skeptical or not, the above is well accepted and is on a solid foundation. There are also various other formulas for pi in terms of infinite series. Such series can be represented finitely. e.g. $$4 \sum_{i=0}^\infty -1^i \frac{1}{2i+1}$$That is a finite representation of an infinite series which yields an irrational result.

Note that if you accept the idea of a "language" at its most general: an arbitrary function from a set of codes to a set of values, there is no problem coming up with a language such that f("this is pi") = $\pi$.

 

[Delta2]

I don't mean those kind of representations, they are like cheating.

Hold on while I make my point about strings of all possible lengths, therefore some strings infinite...

 

[nuuskur]

Is it allowed to quotient the class of all human thoughts with respect to logical equivalence?

 

[Delta2]

Hold on while I make my point about strings of all possible lengths, therefore some strings infinite...

Yes well it all boils down to the statement " The natural (or the real) numbers are infinitely many but each number is finite". This might be true for mainstream math but according to my view it hides a contradiction. But since I am not allowed to post against mainstream science I can't expand on this.

 

[jbriggs444]

Yes well it all boils down to the statement " The natural (or the real) numbers are infinitely many but each number is finite". This might be true for mainstream math but according to my view it hides a contradiction. But since I am not allowed to post against mainstream science I can't expand on this.

Since you are biting your tongue and not espousing your non-mainstream viewpoint, I will likewise avoid expounding on the standard viewpoint.

If carefully phrased in a non-confrontational manner, this could be the start to an interesting thread in the mathematics sub-forum. How do we axiomatize the integers? Can there be more than one model which fits the axioms? If so, then those axioms aren't really a definition, are they? Is there a set of axioms that completely characterizes the standard integers.? What is a "model" anyway? How do we judge whether two models are equivalent? I am not smart enough to do it myself, but there may be enough here for an insights article.

 

[Delta2]

Is it allowed to quotient the class of all human thoughts with respect to logical equivalence?

yes sure but that won't get you as far as you think, for example there are thoughts like "abcdfhegj +-kagaluga" which are logically equivalent only to their self as they are complete nonsense..

 

[jbriggs444]

yes sure but that won't get you as far as you think, for example there are thoughts like "abcdfhegj +-kagaluga" which are logically equivalent only to their self as they are complete nonsense..

How do you know that they are complete nonsense without an idea of what a thought is, a way to do a quotient operation on them, or a way to associate them with something external.

This whole discussion as it pertains to "thoughts" is utterly premature without such an underpinning.

We need a measurement process in order to make this scientific.

If we squint our eyes a little, some of this is well trodden ground. It is called the school system. A thought is that which is embedded into a person by the act of "teaching" or "learning". We measure thoughts by things known as "tests".

 

[pbuk]

If carefully phrased in a non-confrontational manner, this could be the start to an interesting thread in the mathematics sub-forum.

I'm not so sure, the question would be something like "please explain the whole of set theory as it applies to the natural numbers and all possible extensions of them; include not just Zermelo-Fraenkel set theory but also explain both the theoretical existence and practical examples of other consistent formulations".

There is nothing to discuss here, the answers are well known but understanding them is about a quarter of each of the first two years of a degree in mathematics. The course I linked at the end looks like a good start.

How do we axiomatize the integers?

However we want to, providing that once we have done it we come up with something that behaves the same way as the things everybody agrees are integers.

Can there be more than one model which fits the axioms? If so, then those axioms aren't really a definition, are they?

This is semantics: the words "model" and "definition" can mean different things to different people. To explore what they mean to you consider the example of the continuum hypothesis (CH): the model where CH is true is consistent with (fits) the axioms of ZFC and so is the model where CH is false.

Is there a set of axioms that completely characterizes the standard integers.? What is a "model" anyway? How do we judge whether two models are equivalent? I am not smart enough to do it myself, but there may be enough here for an insights article.

There is too much for an insights article, what is needed is an introductory course on set theory: this one looks OK.

 

[pbuk]

Yes well it all boils down to the statement " The natural (or the real) numbers are infinitely many but each number is finite". This might be true for mainstream math but according to my view it hides a contradiction. But since I am not allowed to post against mainstream science I can't expand on this.

There is more to "mainstream math" (whatever that is) than the natural numbers, consider for instance the cardinal numbers. But no consistent definition of the natural numbers can include $\infty$ because, for example, every natural number must have an additive inverse.

I strongly recommend that instead of searching inside your current knowledge for apparent contradictions you learn about how we have solved these, and other, puzzles over the past 150 years, and also learn that there are even more interesting unsolved puzzles and, perhaps most interesting of all, that there are some contradictions that we just have to live with!