A Mind-Boggling Number Comparison

[pinball1970]

There is a simpler sequence but along the same lines as the above. The rules seem a little specific and weird but they are easier to follow than TREE (3) you can do it yourself (to a point) and they produce some counterintuitive results.

Very large numbers reached quickly like TREE (3) but what happens if you keep repeating is the interesting part.

How they work out the end results, like TREE (3) and Grahams number is out of reach (for me and non mathematicians) but you can still follow rules of the game.

 

[pinball1970]

So what about TREE (n > 3)? Does that make sense at all?

It is bigger than TREE (3)!

 

[sbrothy]

I really want to put that lightbulb under your post, but if I really don't understand what's going on I would be lying. So I'll give it a thumbs up instead!

 

[pinball1970]

That was what intirgued me about Graham's Number. Tree(3) is big, but is it useful?

Some mathematics is useful outside of mathematics, Engineering, Economics, physics for eg some has no practical use besides making mathematicians very excited!

 

[sbrothy]

I agree that's where math is at it's most beautiful. When it has no application at all. Like String Theory! (I'm joking!)

 

[BWV]

I find graham’s number more interesting as you can mentally go in a sequence from comprehensibly large numbers then get to a point, somewhere between 3^^^3 and 3^^^^3 (using ^ for Knuth up arrows) where your mind stops comprehending. Also, as a power of 3, Grahams number is more practically computable - the last 10 digits are
2464195387
You can see the last 1.6 million digits here

https://ankokudan.org/d/dl/others/Graham-16M.pdf

Less is known about tree(3) - don’t even know if it is even or odd

 

[pinball1970]

I agree that's where math is at it's most beautiful. When it has no application at all. Like String Theory! (I'm joking!)

Some mathematics were in that category but became useful in physics. Hilbert spaces, Riemann geometry, Noether's work on symmetry.
TREE 3 might get us to Andromeda one day ;)

 

[sbrothy]

Some mathematics were in that category but became useful in physics. Hilbert spaces, Riemann geometry, Noether's work on symmetry.
TREE 3 might get us to Andromeda one day ;)

I know. It was a quip.

 

[pinball1970]

I find graham’s number more interesting as you can mentally go in a sequence from comprehensibly large numbers then get to a point, somewhere between 3^^^3 and 3^^^^3 (using ^ for Knuth up arrows) where your mind stops comprehending. Also, as a power of 3, Grahams number is more practically computable - the last 10 digits are
2464195387
You can see the last 1.6 million digits here

https://ankokudan.org/d/dl/others/Graham-16M.pdf

Less is known about tree(3) - don’t even know if it is even or odd

Yes but you can start building the trees and playing the game. With Graham's number how do you start connecting the lines past three dimensions?
You can start to build the number itself using the arrows but how does that relate to vertices?

 

[DaveC426913]

So what about TREE (n > 3)? Does that make sense at all?

Indeed. One can always conceive of a bigger number. Tree(3)+1. ta-da!

Graham's number became known (half a century ago) because it was the first such collossally stupid-big number that was invoked in a serious math paper.

 

[Drakkith]

Just to add, a big number is not particularly interesting, it is how these things are noted, can grow and are used in proofs.

Indeed, it is WHY the number is so big that is interesting.

 

[WWGD]

Want something that grows fast? The number of Latin Squares: nxn matrices where numbers from 1 through n appear exactly once in each row and each column . For n=9, there are 5,524,751,496,156,892,842,531,225,600(For n>1, Sudokus are a proper subset of Latin Squares), out of which an estimated$6.67 \times 10^{21}$ are Sudokus.

 

[WWGD]

One wing of the hotel a holds a Graham’s number of copies of the library, a single volume in each room of the wing placed beside Gideon’s Bible

Did Gideon donate it?

 

[pinball1970]

Indeed. One can always conceive of a bigger number. Tree(3)+1. ta-da!

Not very creative or gentlemanly though, imagine if two professor's played that game? What sort of rules they would have if they tried to beat each other by coming up with the biggest number?
That is what happened apparently at M.I.T. in 2007. The big number duel.

https://web.mit.edu/arayo/www/bignums.html

The popsci version.

 

[WWGD]

Not very creative or gentlemanly though, imagine if two professor's played that game? What sort of rules they would have if they tried to beat each other by coming up with the biggest number?
That is what happened apparently at M.I.T. in 2007. The big number duel.

https://web.mit.edu/arayo/www/bignums.html

The popsci version.

Depending on how you frame that search for largeness, it may be a futile pursuit. There's no such thing as an all-encompassing set, a largest set, pedantically, by Cantor's Theorem. But I'm frankly not too clear on the topic proposed by the OP.

 

[pinball1970]

Depending on how you frame that search for largeness, it may be a futile pursuit. There's no such thing as an all-encompassing set, a largest set, pedantically, by Cantor's Theorem. But I'm frankly not too clear on the topic proposed by the OP.

Can you not have a set of all sets?

I wrote the above then a distant bell rang, also why would a Mathematics poster say there isn't a largest set?
Checked Google and saw B Russell, you can't. I will read why later as I have to dash, I do remember the liars paradox though.

 

[WWGD]

Can you not have a set of all sets?

I wrote the above then a distant bell rang, also why would a Mathematics poster say there isn't a largest set?
Checked Google and saw B Russell, you can't. I will read why later as I have to dash, I do remember the liars paradox though.

No, givem.a set S. you can define its powerset P(S)( the set of all of its subsets), which Cantor Theorem shows has a greater cardinality than S itself. Let |S| denote the cardinality(size)* of S, and $2^S$ denote its powerset, as defined above. A theorem.states: $|2^S|>|S|$. Thus whatever set S you have, you can define its powerset $2^S$, and the latter will have more elements than the former. I hope I explained it clearly.

 

[jbriggs444]

Can you not have a set of all sets?

Given a set of all sets, one can define the set of all sets not containing themselves. Then you ask yourself, does this set contain itself. That is Russell's paradox.

In ZF set theory, one avoids the paradox by replacing the axiom of unbounded comprehension:

"Given a predicate (a true/false statement about a set element), there is a set containing exactly those elements that satisfy the predicate (make it true)"

with the axiom of bounded comprehension:

"Given a set and a predicate, there is a subset containing exactly those elements that satisfy the predicate"

If the set of all sets exists, then ZF is inconsistent.

 

[WWGD]

Given a set of all sets, one can define the set of all sets not containing themselves. Then you ask yourself, does this set contain itself. That is Russell's paradox.

In ZF set theory, one avoids the paradox by replacing the axiom of unbounded comprehension:

"Given a predicate (a true/false statement about a set element), there is a set containing exactly those elements that satisfy the predicate (make it true)"

with the axiom of bounded comprehension:

"Given a set and a predicate, there is a subset containing exactly those elements that satisfy the predicate"

If the set of all sets exists, then ZF is inconsistent.

Or, by Canto0r's Thm: $|2^S|> |S|$, hence $| 2^{2^S}| >|2^S|....$

 

[pinball1970]

Or, by Canto0r's Thm: $|2^S|> |S|$, hence $| 2^{2^S}| >|2^S|....$

Thanks @WWGD and @jbriggs444. I needed to read some back story on sets and things.

 

[pinball1970]

Given a set of all sets, one can define the set of all sets not containing themselves. Then you ask yourself, does this set contain itself. That is Russell's paradox.

This part bent my brain a little