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I have something(s) that causes me both trouble with (keeping emotionally connected to what I'm trying to express) Λ (expressing myself). So please be nice and ask questions :: εὐχάριστος (Grateful)

Supposing that one is limited to Solipsistic Thought...

If there is no maximum number we could count to, wouldn't the maximum number be the highest as to what's (imaginable or perceptible)? Would that mean that there is a maximum number that's countable ?

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    What's the relation with solipsim ? – Mauro ALLEGRANZA Jul 15 '19 at 7:41
  • There is a "minority" view in the field of philosophy of mathematics called Finitism according to which there is no the (infinite) set of all natural numbers. See also Ultrafinitism. – Mauro ALLEGRANZA Jul 15 '19 at 7:44
  • "If there is no maximum number we could count to, wouldn't the maximum number be the highest as to what's (imaginable or perceptible)?" What does it mean ? If there is no maximum number..., then to speak of "the maximun number" is non-sense. – Mauro ALLEGRANZA Jul 15 '19 at 7:46
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    In addition to Mauro's comments about finitism / ultrafinitism, you might be interested in reading "The Truth about Counting" (brianrotman.wordpress.com/articles/…), written by Brian Rotman in 1997. In it, Rotman discusses "the outer horizon of all counting in this universe" from the point of view of physics and available energy. – Alexis Jul 15 '19 at 9:23
  • Oh, this gives me some reading.. I've little formal education and I'm afraid I'm not sure how to get anything more then basic concepts from those sites at the moment. I'm an artist of sorts; it was just a random thought and I only have a GED. Should I remove this question if it's of no use to anyone? – Μετα Jul 15 '19 at 14:00
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First, we must define what we mean by "a number we could count to". If we mean the largest number that we could count to in practice, in one lifetime, then yes, there would be a maximum such number. Even if we skip numbers while counting, there exist numbers large enough that merely saying their name would take longer than a human lifespan. For instance, even if we could pronounce (or think) one digit per millisecond, saying the first 10^15 digits of pi would take 30,000 years.

But that's just due to physical limitations of humans; it's not interesting mathematically. So let's define counting a bit more rigorously. We define the successor function S(n) = n + 1. We then start counting at 0, and the successor function gives us the next number we should count. For any natural number (non-negative integer) i, if we count like this long enough, we will eventually reach i, no matter how large.

But we will never reach a point where we've counted all the natural numbers. If we've been counting one number per second, then after i seconds, we'll be up to the number (i+1). But we can't stop yet; we haven't even covered (i+2), much less (i+42). We can count to any specific natural number in a finite amount of time. But we'd need an infinite amount of time to count all of them.

  • Ahhh ok.. I see how my thinking/thoughts were not in bounds within the topic,, that's usually my problem -_-. (I had a terrible time in school). Along with the concept of :: (in practice : limited) vs (mathematically : unlimited) :: ( Something I've been wondering since the epoch of this discussion is whether we should ever think of anything as infinite, but rather 'there will always be enough for what one needs' rather then the psychological impact of having the concept of 'unlimited' of anything bleed into our selfish desires (which is (maybe?) self reinforcing).) – Μετα Jul 26 '19 at 16:13
  • @Μετα Generally when we see infinities in practice, it isn't that we have an infinite number of some concrete thing. Rather, it will be a property of some function we're looking at, usually as a limit. For example, the reason we can't go faster than light is because the amount of energy needed to go faster increases the faster we're currently going. Specifically, it's proportional to $1/sqrt(1-(v/c)^2)$. As our velocity v approaches the speed of light c, the energy needed keeps increasing. No matter how much energy you have, there's some v where it isn't enough. The limit equals infinity. – Ray Jul 27 '19 at 19:57
  • @Μετα If you want to look into infinities further, the first step would be looking into some introductory calculus so as to understand limits. In most cases where infinity shows up, as in the example above, it isn't a value, but rather the limit of some function or process. Understanding exactly what that means will help a lot. The next big part is understanding cardinality, the different sizes of infinity. I discuss that a bit in this answer. – Ray Jul 27 '19 at 20:05

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