Permalink Submitted by Anonymous on September 9, 2014

I can fully agree with the view that infinity is essentially the lack of boundary. However, the conclusion drawn from this in the previous post is nothing but a pseudo-philosophical game of words. Here is an even worse example: "existence is the non-existence of non-existence". You get nowhere with this.
If one looks carefully at the mathematical practice of set theory, it may be noticed that the members of a sets are rarely "being collected"; only very small finite sets arise like this. Instead, one considers a defining property of the elements to be collected. When presented a potential element, one can (in principle) decide whether it satisfies the property or not. In programming, this is called "lazy construction". As all mathematical thinking is evidently finite, one will never face "actual infinity", but just finite formulations of properties.
In regard of Cantor's result, there are more subsets of the natural numbers than one can formulate properties. So there must be subsets without a defining property. Not surprisingly, little can be said about them (except that they must be infinite). They are like most of the transcendental numbers: vague generic objects filling the container. It should not be surprising that hard questions may arise with such objects, like the existence of sets with cardinalities in between countable and continuum.

## Infinity: a more realistic view

I can fully agree with the view that infinity is essentially the lack of boundary. However, the conclusion drawn from this in the previous post is nothing but a pseudo-philosophical game of words. Here is an even worse example: "existence is the non-existence of non-existence". You get nowhere with this.

If one looks carefully at the mathematical practice of set theory, it may be noticed that the members of a sets are rarely "being collected"; only very small finite sets arise like this. Instead, one considers a defining property of the elements to be collected. When presented a potential element, one can (in principle) decide whether it satisfies the property or not. In programming, this is called "lazy construction". As all mathematical thinking is evidently finite, one will never face "actual infinity", but just finite formulations of properties.

In regard of Cantor's result, there are more subsets of the natural numbers than one can formulate properties. So there must be subsets without a defining property. Not surprisingly, little can be said about them (except that they must be infinite). They are like most of the transcendental numbers: vague generic objects filling the container. It should not be surprising that hard questions may arise with such objects, like the existence of sets with cardinalities in between countable and continuum.