Who intuits that 10 is closer to infinity than 5?

To say nothing of the reals!

Seriously though some of the people here seem to think of infinity as some boundary, or some kind of a Cauchy sequence. and it is neither; numbers really do go on forever. And certainly some infinities are larger than others.

OK I see. The necessary and sufficient (OTO&ONTO) is covered by the defining function. In that case I see how the sets are the same cardinality. Using the same logic you can define the rationals as a function of integers that would be OTO&ONTO--consequently the cardinality is equivalent

Rap

A concept of infinity is the size of the universe. There are no boundaries

Is 5 or 10 closer to the size of the universe?

Dadpad wrote:


Ten is, by my way of thinking.

Raprap-I actually understood your first explanation. I think I understand part of the rest of it, but the concept of intuiting the concept infinity and the vastness of the universe appeals to me more than trying to pin it down by equating it (which it seems can't be done anyway)...

If i had ten sheep, I'd be closer to fulfilling my dream of having infinite sheep than if I had five sheep.

Yes, this is the way I thought of it, Infinity is the Universe.

So I translated the question to be:

Is Location-A any closer to the Universe than Location-B. And that question becomes irrational, so the same must be true of the mathematical question.

That's a nice way of think about it, alternatively always re-structure a general question about infinity to ask can a property of a bounded space be asked of an unbounded space?

So questions like is the square root of a unbounded space smaller or really any different than the unbounded space itself - quickly becomes meaningless when you ask it that way!

Anything that wants you to consider infintities as if there had properties of finite things is likely to be a trick question.

Are you sure the universe is infinite?

Markr - if the poster is certain of that, his / her Nobel prize is secure.

One thing we do know is that the universe must be either expanding or contracting - stationarity being a theoretical impossibility - and through dint of observation cosmologists assure us it's expanding.

There are 3 distinct fallacies implicit in this post - no time to explain in detail now, but please see Whitehead and Russell's Principia Mathematica (specifically what became known as "Russell's paradox", even though a Cretan came up with it some 25 centuries ago) and the work of the great Georg Cantor, in particular his correspondence on the subject. Will provide details later today if able.

I dont think anyone "intuits" that 5 and 10 are equidistant from infinity. Intuitively everyone (with exceptions ofcourse) thinks 10 is closer, only through study and having it impressed opun the individual, does one come to realise it is not the case that 10 is closer.

I'm seein' this both ways. I was thinking about this before I saw this thread (coincidentally?!).

What I was actually thinking about was a line with zero at one end and infinity at the other. I was thinking about asking people to mark where on this line one million would be...knowing that the correct answer would be to mark that point at (or at least infinitely close to) zero.

But "infinitely close to" is not quite the same as equal to. There is an infinitely small difference, no?

Infinity doesn't own a position on the number line. It's not a number.

You mean it's not the arrowhead on either end of the real number line?

Rap

The whole point of this thread was to highlight common mis-conceptions.

1. Infinity isn't a number or member of the positive or negative real or imaginery numbers

2. Infinity isn't on any number line

3. You generally can't do any field based numerical operations on infinity because they work on numbers and infinity isn't a number

4. Infinity is the sense of no boundaries or limits.

So when asked what is bigger infinity or 1 plus infinity, always transalte this to:

What is numercially larger - the sense of no boundaries or the sense of no boundaries plus one?

You'll quickly get the hang of - oh some poor fool is mixing mathematics involving numbers with those involving infinities and believe the same rules and operations apply - they don't!

So next time someone asks what is bigger 2 or cosine infinity - smile and say well ones a numbers and ones a function with a range between -1 and +1 when you give it a number as an input, but infinity is not a number so its like asking what is bigger 100 or the square root of red?

Very cool g_day. Thankyou!

Technically, you could put infinity on a number line. You'd just have to have a "..." in the middle. A break in it. And then you would see that 10 is closer to infinity than 5 is.

If by "technically" you mean non-mathematically, then you're right - anything goes.

Infinity is four blonds in four cars at a four way stop ..... soooo, I intuit five.

By technically, I don't mean non-mathematically. I've had math teachers put infinity on a number line.