Robert J. Marks: A few examples will show the absurd results that come from assuming that infinity exists in the world around us as it does in math:

Cantor’s theory of the infinite can be explained, starting with the lowly shepherd tending sheep. Imagine a shepherd who does not count well. He gathers stones until the number of stones is equal to the number of sheep he is tending. The set of stones is said to have the same size, or cardinality, as the set of sheep. If there are ten sheep, the size of the set of sheep is ten and the shepherd picks up ten stones. The size of the set of sheep is the same as the size of the set of stones because there is a one-to-one correspondence. At the end of the day, the shepherd compares the number of stones to the number of sheep. If the number of the stones is the same as the number of sheep, no sheep have been lost. Sheep #1 corresponds to stone 1, sheep #2 to stone 2, sheep #3 to stone 3 all the way up to the tenth sheep…

A set with a true infinite size is the set of counting numbers.

{ 1,2,3,4,…}.

The infinite size of this set is said to be the Hebrew letter aleph: . Let’s play around. Take away the first number in this set, namely 1, to get the set

{2,3,4,5,…}.

Even though we’ve made the original set smaller in one sense, both sets have the same size. Think of sheep and stones. Sheep number 1 now maps to stone #2. Sheep number 2 maps to stone #3. Sheep 3 to stone #4 etc. The sets never end so this correspondence goes on forever. There is a one-to-one correspondence between the elements of the two infinite sets so the size of both sets is identical.

This result should strike you as ludicrous. The infinite size of the sets might be the same, but the diminished set is clearly smaller than the set of counting numbers in another sense. This is weird. Let’s apply this same simple idea to the infinite. We’ll show that doing so leads to ludicrous conclusions. Such conclusions are common when dealing with infinities.

Robert J. Marks, “1. Why Infinity Does Not Exist in Reality” atMind Matters News

But this is nothing compared to what happens when Dr. Marks gets to Hilbert’s Hotel …

*Takehome:* Robert J. Marks: In a series of five posts, I explain the difference between what infinity means — and doesn’t mean — as a concept.

*Next:* Infinity illustrates that the universe has a beginning. The age of the universe is shown to necessarily be finite if ridiculous properties of infinity are to be avoided.

*You may also wish to read:* Yes, you can manipulate infinity in math. The hyperreals are bigger (and smaller) than your average number — and better! *(Jonathan Bartlett)*