Philosophy Now Forum

An integer=a non-integer: Mathematics ends in contradiction
http://gamahucherpress.yellowgum.com/wp ... ssible.pdf
very simple
1 is a finite number it stops
A finite decimal is one that stops, like 0.157
A non-finite decimal like 0.999... does not stop
when a finite number 1 = a non-finite number 0.999.. then maths ends in contradiction

another way
1 is an integer a whole number
0.888... is a non-integer it is not a whole number
0.999... is a non-integer not a whole number
when a integer 1 =a non-integer 0.999... maths ends in contradiction

I all fairness this one is solvable by abandoning infinitism and embracing decidability.
The reason for this happening is because x = x is assumed true, not asserted true.

The difference between assumption assertion is simply - work. You HAVE to do work in order to assert equality.

Demonstration:

Observe how the following takes almost no time to evaluate: 1 = 1 => True
Observe how the following takes a little longer to evaluate: 88888888888888888881 = 8888888888888888881 => True

Evaluating the equality of an infinite number will take you an infinite amount of time e.g x = x is undeterminate for infinities.

Now also observe that 88888888888888888881 = 8888888888888888881 => True is an error which becomes far easier to see when you present it as follows:

88888888888888888881
8888888888888888881

The length of the number determines the complexity of evaluating: x = x.

It's known as the range-precision trade-off in FINITE floating point arithmetic.

https://en.wikipedia.org/wiki/Floating-point_arithmetic