Who intuits that 10 is closer to infinity than 5?

raprap wrote:

However, in my mind a logic exists. You could partially prove the theorem using two sets, one set (A) containing all numbers (natural or otherwise) from 5 to positive infinity (inclusive?) and the other (B) all numbers from 10 to positive infinity. The first set (A) then contains numbers that are not in the second one (B), whilst the intersection of the two sets (A intersection B=B) is the same as the second set (B). Consequently, there is no one to one relationship from the elements in A to B and the sets do not have the same cardinality (number of elements).

So set A is larger than B. QED(?)

Rap

No. There is a one-to-one mapping of set B to set A:
b -> b-5
The sets have the same cardinality. Your proof doesn't apply to infinite sets.

Is infinity closer to infinity than 5? Or 10 if you prefer.

spendius wrote:

Is infinity closer to infinity than 5? Or 10 if you prefer.

Woldn't infinity = infinity and thus be equally close to 5 or 10.

Consequently, there is no one to one relationship from the elements in A to B and the sets do not have the same cardinality (number of elements).

2) markr, raprap's logic is correct...in that looking for a one-to-one and onto mapping is the preferred way of considering the size of an infinite set...although their cardinality is the same, this does show that B is a subset of A.

5) markr, yes there is a one-to-one mapping, but raprap meant to say one-to-one AND onto...and it's not onto.

maporsche wrote:

spendius wrote:

Is infinity closer to infinity than 5? Or 10 if you prefer.

Woldn't infinity = infinity and thus be equally close to 5 or 10.

If A = infinity and B = infinity, A does not equal B unless you know for some outside reason that A = B. Read the existing posts...

Cruising at 2 to the power infinity -1, (Ed. note - a special number that only has meaning in the field of improbability physics), Douglas Adams - the Hitch Hikers Guide to the Galaxy.

Guys this Infinity -1, squared, square root etc are all examples of field operations on numbers, the number line example above is the classical set up that gives folks head aches latter on.

In limits you can play with infinities carefully, in non Cartesian geometry where you have a horizon at infinity you can play with infinities, in integration you canplay with infinities, in set groupings and relative sizings you can play with infinities, but in all other places you shouldn't because you can't!

Infinity - 1 is a no sensiblle concept that maps to any meaningful number. Infinity maps to no sensible number. You can't do numerical operations on infinity like you can on numbers and arrive at any sensible conclusion, its Verbotten!

Also from memory, 1/0 = undefined, commonally thought of as equally all numbers simultenaously, so its one more example of where numbers and infinity don't mix.

Even 0 * infinity isn't well defined. The zero of a field times the concept (that isn't in the field) but is instead the idea that the field is not bounded, is non sensible. You can't multiply the lack of a boundary of a field by the zero generator of the field and get a number!

There is no such thing as an infinite "number" - that's an oxymoron. There are infinities, there are numbers, the two are distinct and different. One is a concept of no limits, another is a specific number in a field of numbers.

Infinities aren't numbers. You can't multiply zero by the lack of a boundary; zero only works on numbers.

It's like asking what's three times red - you can't sensibly multiple a colour by a number either.

f(b) = b-5 is one-to-one AND onto with respect to sets A and B.

For every a in A, there is a b in B such that f(b) = a

BTW this discussion rekindled a memory of a similar problem from my dimming past--a discussion of the equivalence of natural numbers and positive integers. If memory serves me well (& many times it doesn't) the sets aren't equivalent, mostly because one to one mapping function is not onto (natural number mapping onto positive integers isn't unique as |+/-a|=+a)

Maybe I'm making it too simple.....but on a number line

Code:

0---1---2---3---4---5---...---Inf



It would seem that 5 is closer to infinity than 1 is. Another way to look at it would be is Infinity minus 5 less than Inifinity minus 1.

maporsche wrote:

Maybe I'm making it too simple.....but on a number line

Code:

0---1---2---3---4---5---...---Inf



It would seem that 5 is closer to infinity than 1 is. Another way to look at it would be is Infinity minus 5 less than Inifinity minus 1.

since nobody responded to this, i can. :wink:

the whole point of infinity is that you can't put it on a number line. otherwise, you could put a number to the right of it, like infinity + 1, which would be larger than infinity, but that contradicts the definition of infinity.

yitwail wrote:

maporsche wrote:

Maybe I'm making it too simple.....but on a number line

Code:

0---1---2---3---4---5---...---Inf



It would seem that 5 is closer to infinity than 1 is. Another way to look at it would be is Infinity minus 5 less than Inifinity minus 1.

since nobody responded to this, i can. :wink:

the whole point of infinity is that you can't put it on a number line. otherwise, you could put a number to the right of it, like infinity + 1, which would be larger than infinity, but that contradicts the definition of infinity.

It would make sense that nothing can be greater than infinity, but I don't understand why there are problems with something being varying degrees less than infinity. Inf - 1 vs Inf - 10,000 for example.