Let's Talk About Infinity

[History_Buff]

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.

All of this. Obvious infinity exists.

And to address the OP, I know this to be true, yet it certainly does not inspire a faith in a god.

 

[Ramius75]

The only impossible thing is finite time and space... as what is before the beginning and whats after it ends?

 

[lovett]

Some confusion to clear up to begin with.

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.

It doesn't matter what type of world we are living in. Our understanding of what 'theoretically possible' means can be tested through hypothetical; this is what the deterministic world case achieves. If a certain understanding of 'theoretically possible' fails in such hypotheticals, it is a misunderstanding.

So, for instance, knowledge has been analyses as 'justified true belief'. However, we can imagine hypothetical cases in which one has JBT but not knowledge. For alacrity, I'll paste one from wikipedia:

"After arranging to meet with Mark for help with homework, Luke arrives at the appointed time and place. Walking into Mark's office Luke clearly sees Mark at his desk; Luke immediately forms the belief 'Mark is in the room. He can help me with my logic homework'. Luke is justified in his belief; he clearly sees Mark at his desk. In fact, it's not Mark that Luke saw; it was a marvelous hologram, perfect in every respect, giving the appearance of Mark diligently grading papers at his desk. Nevertheless, Mark is in the room; he is crouched under his desk reading Frege. Luke's belief that Mark is in the room is true (he is in the room, under his desk) and justified (Mark's hologram is giving the appearance of Mark hard at work)."

The situation above never happened, but we can nonetheless use it to test our concept of knowledge. It turns out, knowledge can't be just JBT. Similarly, we can test our understanding of 'theoretically possible' via hypothetical.

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.

Of course (to the bolded part). But I am at a loss as what you mean to derive from this. Claims regularly need to be understand in the language of possible worlds (counterfactual claims are analysed this way. claims of the form; if X were there case Y would be the case'). Just because possible worlds are not the actual world does not mean our concepts can avoid referencing possible worlds in their analysis. Certainly, any plausible theory of knowledge relies on counterfactual claims and thereby possible worlds.

Indeed, that's what I said

Are you sure? You said:

"It's the system of measurement itself which is infinite, in the sense that any unit of counting can be broken down into an infinite number of smaller sub-units. "

I am disputing this. I am saying that systems of measurement refer. They refer to certain lengths. If those certain lengths do not exist (even theoretically) one cannot refer to those lengths. Hence, the system of measurement would 'stop'. One could still do things with the numbers, but those numbers would no longer constitute a system of measurement, precisely because they no longer referred to a certain length.

I am arguing that this situation obtains when we try and go below the Planck length. If it is the smallest possible length, no length below it is smaller. Hence, no system of measurement could refer to 'Planck length/ 2'.

Perhaps I can make this clear with a (better) analogy. Suppose I wish to develop a specific system of measurement for measuring the comparative size (in terms of population) of countries. I am interested in only one aspect of this comparative size; which countries are bigger than which other countries. This is perfectly valid; I do not need to be interested in actual population figures to derive a system of measurement.

My system is numeric. Namely, I take the biggest country first and proceed downwards.

So, '1' refers 'China', '2' to 'India, '3' to 'The USA', '4' to 'Indonesia', so on and so forth. In the same way, measuring lengths '1 metre' refers to a certain length (100 centimetrea), '2 metres' another length (200 centimetres) so on and so forth. On my new system of measurement, I can look up '27' and find what it refers to.

Being numberic, I can divide the numbers I use in the system of measurement infinitely. I can multiply them infinitely. But, I clearly cannot do so for the system of measurement itself. I cannot expand the system past '192' nor divide any number the system by more than 96. The actual system is finite, because it refers to real finite things. The numbers are not, because of the structure of mathematics.

Similarly, I am arguing that in reality length is finitely divisible. That means systems of measurement of length are finitely divisible. If one attempted to infinitely divide, one loses the ability to refer to lengths.

 

[Leoreth]

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

What makes you think that?

But we don't know if we are living in a deterministic Universe or not.

We're pretty sure that our universe is not deterministic.

- there are the same number of rational numbers as integers, despite the fact that there are infinitely many rational numbers between each integer

Minor nitpick, but "the same number" might be a misleading term in this context.

The only impossible thing is finite time and space... as what is before the beginning and whats after it ends?

What's south of the south pole?

 

[Narz]

So, for instance, knowledge has been analyses as 'justified true belief'. However, we can imagine hypothetical cases in which one has JBT but not knowledge. For alacrity, I'll paste one from wikipedia:

"After arranging to meet with Mark for help with homework, Luke arrives at the appointed time and place. Walking into Mark's office Luke clearly sees Mark at his desk; Luke immediately forms the belief 'Mark is in the room. He can help me with my logic homework'. Luke is justified in his belief; he clearly sees Mark at his desk. In fact, it's not Mark that Luke saw; it was a marvelous hologram, perfect in every respect, giving the appearance of Mark diligently grading papers at his desk. Nevertheless, Mark is in the room; he is crouched under his desk reading Frege. Luke's belief that Mark is in the room is true (he is in the room, under his desk) and justified (Mark's hologram is giving the appearance of Mark hard at work)."

Man, I hate when that happens! Makes me feel like such a fool. That damn Mark!