School of Mathematics and Statistics, University of New South Wales.
Norman correctly points out, that the idea of a completed infinite sequence of numbers is indeed nonsense.
The point stated above is covered several times throughout this website, but it’s worth mentioning here – this is exactly how the misconception of infinity manages to manifest into becoming what some people refer to as the “delusion” – something cannot be both infinite and completed at the same time – such a suggestion is a fundamental misunderstanding of what an infinite state is supposed to be!
“Gauss, before Cantor’s time said, (“I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.”)”
“The great French mathematician Henri Poincaré stated (“There is no actual infinity, that the Cantorians have forgotten and have been trapped by contradictions.”)”
“Hermann Weyl one of the 20th centuries most influential mathematicians, stated (“... classical logic was abstracted from the mathematics of finite sets and their subsets …. Forgetful of this limited origin, one afterwards mistook that logic for something above and prior to all mathematics, and finally applied it, without justification, to the mathematics of infinite sets. This is the Fall and original sin of [Cantor's] set theory ...”)”
Quote: “we’re not saying there must be an actual infinity of say, atoms, but we are saying that infinity is a completely Kosher number the same as the others, that could in fact be realised, and that we can understand what it means”
Infinity cannot be a “completely kosher number the same as the others”, simply because numbers are finite values, and infinity isn’t a numerical value, and therefore infinity is impossible to realise in this sense – or in any other sense.
A platonic existence of any numerical value must still be that very same number, and it must still correspond to: division, addition, etc. Therefore, no infinite platonic number exists either.
More examples: any finite value * infinity = infinity, therefore, no numerical value is infinitely divisible, i.e., represents an infinite number of divisions which are each above 0 in value – space cannot extend infinitely far, and time had a beginning, otherwise continuity of existence is broken.
“Three-ness” refers to the supposed abstract existence (i.e., independent of physical reality) of quantity, which is asserted to be real based on the fact that three things can and do exist in our reality. One aspect (extension) of this platonic existence is the assertion of quantitative infinite existence, i.e., completed infinities (i.e., contained, e.g., in a set) of different sizes.
A more detailed explanation about: “Platonic numerical paradoxes” – via this link to the “Platonic Infinity “...” Numerically Paradoxical” page.
Quote: “There's one classic straightforward and naive argument that has always been at the heart of belief in large finite and infinite numbers. However many you have; you can always add one more. It's a sound argument, as it relies on an understanding of numbers founded on an experience of small numbers, but to which there being small is not relevant.”
The process of adding 1 to any finite numerical value simply makes the value larger by 1. Therefore, it is indeed correct that both small and large numbers increase in size after adding 1.
For a value to be infinite, it must be that its size remains the same, even after adding 1 to it – e.g., 2 + 1 = 2, but of course in reality 2 + 1 = 3, and therefore 2 is not an infinite value.
Suggesting that the past might be “potentially infinite” (a misnomer) – a fundamental misunderstanding, i.e., past time is actualised, which means it makes no sense to suggest a potentially infinite past – a supposedly infinite past is indeed an actual infinity.
Explanation: there are two options for past time, both which apply exactly the same for both ways of thinking about causality... time flowing... tenseless, i.e., “block universe”...
There was a beginning to the sequence.
There wasn’t a beginning to the sequence.
1st mistake: past time is actualised by default – there is no need to drop a grain of sand along the way, because causality is always occurring. Technically speaking, we can even just simply think of time that has past as being the number (count) of changes that have occurred in the universe.
2nd mistake: if there supposedly exists an infinite number (count) of causal steps in the sequence of time – not only is it meaningless to add to causality by suggesting additional tasks such as dropping sand... but the following paradox exists:
Continuity paradox: also see “Straight-line distance infinity analysis” – any supposed infinitely faraway point in any type of sequence, cannot actually be connected to any other point in that sequence...
Time example: the present which we know is right now, cannot be connected to an infinitely long-ago moment in time, because that would mean a break exists in the continuity of time – an impossibility, hence time did indeed have a beginning.
Distance example: likewise, any point in space that has a position relative to ourselves, cannot be connected to points that are infinitely far away.
Three options for what we might want to consider to be the largest number...
But remember... numbers don't have an existence of their own.
The number of particles in the entire universe.
The largest number ever processed by a computer.
Maybe around 7 or 8... if you're under 2 years old.