# Thread: The axiom of infinity is surely provably FALSE?

1. ## The axiom of infinity is surely provably FALSE?

The axiom of infinity is surely provably FALSE:

asserts the existence of a set containing all the natural numbers’

• There is no number X such that X>all N because X+1>X
• So Actual Infinity does not exist mathematically
• Actual Infinity does not exist in the material world
• So set theory should not assert its existence

2. ## Re: The axiom of infinity is surely provably FALSE?

Set theory, and, in particular, the "axiom of infinity", in asserting the existence of a set containing all the natural numbers, does NOT assert the existence of a number, "infinity". It does assert the existence of "infinite cardinalities" but they are not "numbers" in the usual, algebraic, sense and, in particular, "X+ 1" is not defined for infinite cardinalities.

3. ## Re: The axiom of infinity is surely provably FALSE?

Set theory asserts the existence of the impossible - an actually completed infinite set.

Actual Infinity is a religious/spiritual concept only, it does not exist mathematically or in the material world.

Cantor only included it in his work because he believed god was infinite.

4. ## Re: The axiom of infinity is surely provably FALSE?

Originally Posted by DannyTR
The axiom of infinity is surely provably FALSE:
[FONT="]asserts the existence of a set containing all the natural numbers’[/FONT]
• There is no number X such that X>all N because X+1>X
• So Actual Infinity does not exist mathematically
• Actual Infinity does not exist in the material world
• So set theory should not assert its existence
For anyone who has taught this material, it is difficult to know where you found the above.
This webpage gives the standard set of axioms.
There you will find references that will help you. I recommend the book by Halmos.

5. ## Re: The axiom of infinity is surely provably FALSE?

My point is one of the standard axioms, the ‘axiom of infinity’ is plain wrong.

Actual Infinity does not exist mathematically or materially so set theory should not define it to exist.

Some of calculus might be on dodgy logical ground too for example:

1 + 1/2 + 1/4 + 1/8 ...
<> 2

It tends to two but never reaches 2. Writing it as equals 2 is a logic error.

There is a big logical difference between ‘two’ and ‘just under two’.

6. ## Re: The axiom of infinity is surely provably FALSE?

Originally Posted by DannyTR
My point is one of the standard axioms, the ‘axiom of infinity’ is plain wrong.
Actual Infinity does not exist mathematically or materially so set theory should not define it to exist.
Some of calculus might be on dodgy logical ground too for example:
1 + 1/2 + 1/4 + 1/8 ...<> 2
It tends to two but never reaches 2. Writing it as equals 2 is a logic error.
There is a big logical difference between ‘two’ and ‘just under two’.
You are missing what it means to say $\displaystyle \sum\limits_{k = 0}^\infty {\frac{1}{{{2^k}}}} = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 0}^n {\frac{1}{{{2^k}}}} = 2$

To understand what that means, one needs to understand limits.
To say $\displaystyle \mathop {\lim }\limits_{x \to \infty } f(x) = \infty$ simply means that as $x$ increases without bound then $f(x)$ also increases without bound.

To say $\displaystyle \mathop {\lim }\limits_{x \to \infty } f(x) = L$ simply means that as $x$ increases without bound then $f(x)$ approaches $L$.
Equally in the ordinary sense in not intended.

7. ## Re: The axiom of infinity is surely provably FALSE?

To increase without bound is to encounter a logic error (eg try implementing infinite in a computer).

Define ‘without bound’ mathematically for me please.

8. ## Re: The axiom of infinity is surely provably FALSE?

Oh, I see. What confused me was that this was posted in "mathhelpforum" when, in fact, you don't intend to say anything about "mathematics" but rather about computers and their limitations.

A sequence $\{a_m\}$ increases "without bound" if and only if, for any number, X, there exist N such that $a_N> X$.

9. ## Re: The axiom of infinity is surely provably FALSE?

Not just computers that can’t handle the actually infinite:

- There is no expression for it in maths (it is just defined axiomatically to exist)
- Not constructable geometrically (not enough graph paper)
- It does not exist in the material world
- Our finite minds cannot fully visualise an infinite set

Actual Infinity does not exist and set theory should not say it does.

10. ## Re: The axiom of infinity is surely provably FALSE?

Originally Posted by DannyTR
To increase without bound is to encounter a logic error (eg try implementing infinite in a computer).
Define ‘without bound’ mathematically for me please.
First: computers are in no way a model of reality. To think they are is known as the existential fallacy.
You seem to have a naive view of axiom system. Mathematics is based on the axioms of Zermelo–Fraenkel set theory.
One accepts that axiom system or not. If not then this thread is useless.
To say that $n$ increases without bound simply means that if $N\in\mathbb{N}$ then $n\ge N$.

11. ## Re: The axiom of infinity is surely provably FALSE?

One excepts an axiom system if it is logically sound.

How is it logically sound for a completed actually infinite set to exist?

So we cannot accept the ‘axiom of infinity’.

12. ## Re: The axiom of infinity is surely provably FALSE?

Are you spamming every math help website with this position of yours?

Bringing it to one site for discussion seems appropriate but on MyMathForum the discussion seems to have gone into the weeds with as far as I can tell every person on that site telling you you need to study more mathematics.

So now you bring it here. Most likely to rehash exactly the same stuff that has been gone over in the 100+ post thread you started over there.