Cantors Infinity

[Dubbelosix]

Consider a machine that will spit out numbers in ascending order, designed to do it for all time. That alone is a concept of infinity., but every time you check the machine to see what number it has arrived at, will always be a finite number. So maybe infinity is a really nice thing we like to think about but can never actually be achieved.

 

The hypothetical machine that will spit out a number in ascending order, will never actually reach infinity, not unless it did it for an infinite amount of time without bound, meaning there is no actual point ''in time'' in which infinity is ever calculated.

 

Countable in Cantors theorem just means, in theory or concept, you can count from 1 + 2 + 3 indefinitely, and while it may be unending, it is still ''countable.''

 

There may be a fallacy here: No infinity is ever countable, in fact, only finite numbers can ever be generated (at one point in time). Considering this, the hypothetical machine, an infinity will never be produced will never show up on its screen, because it would take an (infinite amount of time) to get to it. 

 

Therefore, as far as observers are concerned in spacetime, an infinity can never be produced.

 

Even though I have stated here that infinities cannot be produced in nature, then why do we use it in mathematics to the extent that we do? Consider infinity as a limit [math]x \rightarrow \infty[/math] - it means that [math]x[/math] will grow without bound. In the context of the countable infinity, this is like thinking about a system ascending in order [math]1 + 2 + 3... n[/math] indefinitely. Thinking about the limit as ''approaching infinity'' does not make sense either, since the infinity symbol is not even in the same class of objects as the positive numbers. Infinity is not like a number, its a concept of size rather than a concept of how large a number is. 

 

Of course, it doesn't tend to make sense of thinking of infinities with a ''size'' but of course, this is what was shown at least in theory by Cantors sets, showing that not all hypothetical infinities are even of the same magnitude. I only say it doesn't tend to make sense because generally speaking, an infinity is an object without bound, but this can be discussed at a later time. This initial post was just to quickly cover my hypothetical machine that only ever produces finite results in any given one time meaning that for infinities to exist, they have to be infinite in time. There is nothing to suggest in physics that such a system is even capable of existing and the scientist in fact, hates it when singularities arise in the equations, it is usually an indication that we have got something wrong.

 

I'll continue this later, but if anyone wants to contribute to the discussion I welcome it!

 

 

Edited May 8, 2017 by Dubbelosix

[Maine farmer]

 

 

Even though I have stated here that infinities cannot be produced in nature, then why do we use it in mathematics to the extent that we do? Consider infinity as a limit [math]x \rightarrow \infty[/math] - it means that [math]x[/math] will grow without bound. In the context of the countable infinity, this is like thinking about a system ascending in order [math]1 + 2 + 3... n[/math] indefinitely. Thinking about the limit as ''approaching infinity'' does not make sense either, since the infinity symbol is not even in the same class of objects as the positive numbers. Infinity is not like a number, its a concept of size rather than a concept of how large a number is. 

 

This is exactly how my high school calculus teacher explained it, and I often wondered why none of my other instructors taught it that way. 

 

How can finite minds expect that they can truly comprehend the infinite?  How can we say with certainty that the universe is expanding when we can view such a tiny portion of the whole?  I have enjoyed your other posts, even if it forces me to try to work the rust out of the mathematical part of my brain.

[sluggo]

Cantor was a great illusionist.His advantage was the same as Darwins. He knew more about the subject than the masses, so they considered him an expert.

Infinity should be banned from the english language. It's a meaningless term and allows for distorted interpretations. It's NOT a number but a condition or relationship.

How do you approach infinity, when it's sleeping, from behind, or on your tip toes? You can't any more than you can approach the horizon. Mathematicians should have known better when they defined a limit!

Edited May 12, 2017 by sluggo

[phillip1882]

the goal of countably infinite defined by cantor is to differentiate that from the uncountably infinite of the real numbers.

 

that is there is a 1->1 mapping of the integers to the rational numbers, but no 1->1 mapping of the integers to the irrational numbers.

yes we cannot truly conceive of infinite, but there are different levels of infinite. that was cantors point.