Let's Talk About Infinity

[Save_Ferris]

Split off from here

Sorry, guys, but you were flooding my thread.

Anyway, does infinity exist? Is it possible for something to be infinitely big or infinitely small? Is time infinite? I personally believe that the difference between theists and atheists is the belief of the infinite or finite, respectively.

 

[Ayn Rand]

'Theoretically' takes on odd dimensions here, doesn't it?

Usually we use theoretically to mean 'in principle' possibility. So, humanity can theoretically visit Alpha Centauri because it is 'in principle' possible to do so. However, suppose we live in a deterministic universe in which humanity never visits Alpha Centauri. Is it still theoretically possible to do so?

But we don't know if we are living in a deterministic Universe or not. And even if we are, we don't know the future that it holds.

Unknown laws of reality [such as determinism] and unknown futures are of no application to the concept "theoretical". Theoretical means that which is known, or considered to be known.

Otherwise, we would not be dealing with whether it is theoretically possible, but whether it is going to happen in fact.


As for the rest of your post, you do realise there are an infinite number of counterfactual, possible Worlds? And only one real World. So the theoretical deals only with the one World that exists, else it is the imaginary. And there must be some combination of knowledge and uncertainty [to form the "theoretically possible"], else it is the factual.

Refer to my reply to Sill. The system of measurement is meant to be one which refers to actual lengths. It only refers to those lengths as long as those lengths actually or hypothetically exist. It no longer refers to those length if they do not actually or hypothetically exist, as is(as I'm understanding the physics) the case with lengths smaller than the Planck length.

Systems of measurements are based on mathematical systems. Namely, the mathematical system of real numbers. Real numbers can be divided (and expanded) infinitely. However, that does not entail that systems of measurement can be divided (or expanded? Interesting) infinitely. This is because, as mentioned, systems of measurements refer.

A rather colourful analogy (and not a particularly close one; don't take it too seriously) might be my ability to write posts in forums. This ability is based on sentences in the english language. Sentences in the English language can be expanded infinitely (through conjunctions), but this certainly does not entail that my ability to write posts on forums can be expanded infinitely; it is starkly limited.

Indeed, that's what I said

 

[ParadigmShifter]

Even if there are an infinite number of counterfactual worlds, none of them will ever take libertarians/objectivists seriously.

 

[Ayn Rand]

Even if there are an infinite number of counterfactual worlds, none of them will ever take libertarians/objectivists seriously.

Why would they? Our philosophy is not designed for an infinite number of counter-factual Worlds, unlike yours.

 

[Narz]

Anyway, does infinity exist?

Does it matter whether it does or not? We are finite.

 

[ParadigmShifter]

Why would they? Our philosophy is not designed for an infinite number of counter-factual Worlds, unlike yours.

Your philosophy is not designed for a finite number of worlds > 0 either

 

[Ayn Rand]

meh, why respond?

 

[ParadigmShifter]

Is that a poor attempt at trolling? Could try harder.

 

[warpus]

Anyway, does infinity exist? Is it possible for something to be infinitely big or infinitely small?

Yep. The set of all integers is infinite.

I personally believe that the difference between theists and atheists is the belief of the infinite or finite, respectively.

But we don't know if we are living in a deterministic Universe or not. And even if we are, we don't know the future that it holds.

Unless there's been a breakthrough in the field I'm not aware of most physicists would agree that the universe is non-deterministic.

 

[ParadigmShifter]

Yep. The set of all integers is infinite.

Countably infinite, which is the smallest infinity of all.

There are (infinitely) more real numbers between 0 and 1 than there are integers.

 

[warpus]

Countably infinite, which is the smallest infinity of all.

There are (infinitely) more real numbers between 0 and 1 than there are integers.

Yep.

But both sets remain infinite.. he was asking for something that's infinite in size, so I provided the first thing that popped into my head.

 

[Integral]

Facts about infinity
- there are many gradations of infinity
- the infinity that most people think about, the infinity of the integers, is indeed the smallest of the infinities
- there are the same number of rational numbers as integers, despite the fact that there are infinitely many rational numbers between each integer
- Between any two rational numbers there is an irrational; and between any two irrationals there is a rational, yet there are more irrationals than rationals.
- there are more real numbers than rational numbers
- there are no infinities "between" that of the integers and that of the reals. Maybe.

"Countable" vs "uncountable" is merely scratching the surface.

 

[Skwink]

I don't know, nobody has ever gone to infinity before.

 

[Blue Emu]

Facts about infinity
- there are many gradations of infinity
- the infinity that most people think about, the infinity of the integers, is indeed the smallest of the infinities
- there are the same number of rational numbers as integers, despite the fact that there are infinitely many rational numbers between each integer
- Between any two rational numbers there is an irrational; and between any two irrationals there is a rational, yet there are more irrationals than rationals.
- there are more real numbers than rational numbers
- there are no infinities "between" that of the integers and that of the reals. Maybe.

"Countable" vs "uncountable" is merely scratching the surface.

Agreed on all points.

One thing worth mentioning about your first point ("- there are many gradations of infinity") is that while we can construct an infinite number of orders of infinity, we have only found an application for three (four?) of them... possibly the first three, although as far as I know there is no proof that the three "useful" infinities are the three lowest-order types.

 

[Theige]

And beyond?

 

[Joecoolyo]

And beyond?

Dammit I wanted to post that

 

[Save_Ferris]

In my mind, there are four:
Infinite points in one segment of a dimension
Infinite segments in that dimension
Infinite dimensions
Infinite copies of those points

 

[Rashiminos]

Facts about infinity
- there are no infinities "between" that of the integers and that of the reals. Maybe.

Not a fact. This conjecture is known as the Continuum Hypothesis.

 

[Ayatollah So]

If the geometry of the universe is flat or open, spacetime is infinite.

Recent observations have shown that, from 7.5 billion years after the Big Bang onwards, the expansion rate of the universe has actually been increasing, commensurate with the Open Universe theory.[citation needed]

Citation still needed, but I've definitely seen that before, in Physics Today (I just can't remember when).

 

[Blue Emu]

In my mind, there are four:
Infinite points in one segment of a dimension
Infinite segments in that dimension
Infinite dimensions
Infinite copies of those points

Case C appears to be aleph-null.
Case A appears to be aleph-1.
Case B and D both sound like c (the infinity of the continuum, which is generally assumed to be the same order of infinity as aleph-2, but I don't think this has ever been proven).