Yeah, actually, that brings to mind the fact that you can subtract from left to right, it’s just a little more complicate.
If I’m subtracting 0.9 from 1.0, I can start by subtracting the leftmost digits (1-0) to get 1. I write that down followed by a decimal point. Next I try to subtract the tenths digits (0-9). I can’t because 9 is greater than 0. So I take one away from the 1 I already wrote down, reducing it to zero, and I now subtract 10-9 for the tenths digits, leaving me with one. Repeat down the line, and each digit of the answer turns out to be 0. 0.000…
It should be noted, though, that infinity MINUS infinity doesn’t work because of the ability to add any non-infinite (and some infinite) value to infinity and get infinity. It’s a indeterminate form.
I’m not sure how to. I believe I explained it as well as I can in post 328, quoted below:
The above makes mention of “normal math.” If it’s your intention not to use “normal math,” then the flaw consists in your not having laid out what mathematical system you are using.
But if you mean for the proof to work for the kind of math the educated non-specialist understands, then the abovequoted lays out a specific flaw in your proof.
Well, I don’t know about the normal rules of math and no final digit. To me this is an artifact of the idea of infinity being endless… a reasonable assumption about an infinite sequence of numbers to make, but perhaps not entirely true.
For if we can move from point A to point B and there are an infinite number of points in that movement, why can I not move to the end of an infinite string of 9s? or 0s?
By the agreed upon definitions that everybody else is using there is not and cannot be any such thing as the infinite’th place. Infinite is defined as endless. It does not matter what size infinity you are talking about. They are all endless. They cannot be put into 1 to 1 correspondence but that is a property that is different than endlessness.
There’s the flaw in your argument. No matter how often you repeat variations of it, they all come down to the same error. You are looking at our definition, throwing it out, and then saying that our answers are wrong.
As many people have patiently explained, it is possible to construct arithmetics that start with a different definition of infinity. You are not doing so. You are simply stating that you don’t like the standard definition and attempting to apply an alternate one without any rigor.
When we say that .999~ represents an unending series, you can never subtract .999~ from .999~ and find a place where the difference is .000~1. (By the definition we are using .999~ is an infinite sequence. I have no idea what you mean when you say it isn’t. The tilda at the end is equivalent to using … at the end. Both are notations that mean infinite, again by definition.) You can’t just want the one to be there. It’s not there. It can’t be there, because by definition the infinite’th place doesn’t exist. Whenever you try to stick an infinite’th place in, the rest of us using the standard definition will point out your error.
The reason we like the standard definition is that it solves the problems that once were intractable, like Zero’s Paradox. (Also, please note that the paradox can be represented in a literally infinite number of ways. 1/2 + 1/4 + 1/8 + … = 1 is just another way of expressing 0.9 + 0.09 + 0.009 + … =1. You can find infinite expansions that grow much more slowly yet equal 1. And expansions can converse on any number. Pi must be calculated by an infinite sum. There are an infinite number of these expansions, I believe, yet they all exactly equal pi. Not approximately equal pi; not come close to pi in the terms we can calculate; exactly equal pi. All this is incredibly useful and all this is derivable only because of the definition of endless.)
If you can’t accept that math is entirely axioms, definitions, and proofs derived from them in a rigorous and non-contradictory way, then you can’t accept math at all. You can change the axioms and definitions and get whole and consistent sets of proofs. Everybody agrees that’s true. But once you choose your axioms and definitions, the rest follows and you don’t get to pick which you like and which you don’t.
By the way in Computer Science we talk about computing infinite strings all the time, despite the fact that a computer never could. I would refer you to Turing Machines.
People seem to want to say we cannot ever reach the end of an infinite sequence, but yet do so to solve Zeno’s Paradox. So I would say that is a contradiction there. You must throw out one or the other, from my perspective?
That it uses the word “Axiom” rather than “Definition” is cold comfort…
Furthermore, I see from your first post that the version of the Axiom of Archimedes you use is “For any positive ɛ there is a finite integer N such that Nɛ > 1”.
This is literally the statement “Every positive ɛ is non-infinitesimal”, or, in other words, “There are no (nonzero) infinitesimals”. It doesn’t use the word “infinitesimal”, but only because it hasn’t been translated in that fashion; the definition of “infinitesimal” is “ɛ is infinitesimal if there is not a finite integer N such that Nɛ > 1”.
So the Axiom of Archimedes is not at all intuitive (or true) if you have a numeric system in mind that makes infinitesimal distinctions. It doesn’t bring anything to the table other than “Let’s agree not to draw infinitesimal distinctions”.
I’m aware, I was pointing out how infinity (aleph null) is different from regular numbers. Since it’s not a single value, “infinitieth” place has no meaning. Which place is that? Whichever place you choose, including the ill-defined “infinitieth” will have another place after it.
Well, yeah, you can trivially make a TM or FSM (or whatever else) that can accept an infinite string, but if you give it a string that it can accept, I can guarantee you it won’t halt (for instance, an FSM with two states, a starting state acceptor with a self-loop on a, and a transition to a rejecting state on any other input will never halt given an infinite string of a’s). I’m not sure what your point is. In comp sci we talk about infinite strings, in math we do too. I’m not sure what you’re trying to say.
My comment about computer science was in hindsight irrevlivent.
okay fair enough you point about another place after it as well.
So how do reconcile the anology i am making to Zeno’S Parodox then? We progress through an infinite number of halves:
1/2 + 1/4 + 1/8 + … + 1/infinity to get from point A to point B
I am relating each of these points to numeral in the number .999… or 1.000… how is this a different infinity?
You’re making the mistake of writing 1/infinity in the first place. In the Real number system, 1/infinity is undefined.
1/2 + 1/4 + 1/8 …
There is no 1/infinity, only a series of smaller and smaller points, forever and ever, in perpetuity. Whichever numbered term you pick, there’s another one after it. That’s literally the definition of infinite.
