Cantor’s Paradise

God was with the righteous at his right hand and felt an emptiness at his left. It is not good for us to be bored. Let us make Cantor’s Paradise.

And He looked to his left and saw nothing
$$\mskip -100mu \emptyset.$$
And He called it \(0\) and a day and a night passed.

He looked again the next day and saw
$$\mskip-100mu \{\emptyset\}.$$
And He called it \(1\) and another day and another night passed.

He looked again the next day and saw
$$\mskip-100mu\{\emptyset, \{\emptyset\}\}.$$ And He called it \(2\) and another day and another night passed.

He looked again the next day and saw
$$\mskip-50mu \{\emptyset,\,\{\emptyset\},\, \{\emptyset, \{\emptyset\}\}\,\}.$$ And He called it \(3\) and another day and another night passed.

He looked again the next day and saw
$$\mskip-20mu \{\emptyset,\,\{\emptyset\},\, \{\emptyset, \{\emptyset\}\},\,
\{\emptyset,\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}\,\}.$$ And He called it \(4\) and another day and another night passed.

$$\dots$$

 

 

And God looked upon every thing that he had made, and behold, it was truly good. And Kronecker was happy.

He had created the natural numbers, and calling Kronecker to a rock, he said to him:

—This is not finished, look

$$\mskip-200mu \{\emptyset,\,\{\emptyset\},\, \{\emptyset, \{\emptyset\}\},\,
\{\emptyset,\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}, \dots\}.$$
And He called it \(\omega_0\) and another day and another night passed.

And the next day, he looked again
$$\mskip-200mu \{\emptyset,\,\{\emptyset\},\, \{\emptyset, \{\emptyset\}\},\,
\{\emptyset,\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\}, \dots, \omega_0\}.$$
And He called it \(\omega_0+1\) and another day and another night passed.

And God observed that \(1+\omega_0=\omega_0\ne \omega_0+1\).

Many days passed, many, countless days. And still God continued his work. On the first uncountable day he was able to see an unbelievable set
$$\mskip-200mu\omega_1=\{\emptyset,\,\{\emptyset\},\, \{\emptyset,
\{\emptyset\}\},\, \{\emptyset,\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\},
\dots, \omega_0, \dots\},$$ that no mortal can ever truly know.

And Cantor was happy.

God looked upon his work, and called Cantor to a rock and said to him:

GOD: These are the ordinals. They will be the cornerstone on which I will build your Paradise.

CANTOR: When will I be able to see it?

GOD: As soon as I finish with the ordinals.

CANTOR: Can you do it?

GOD: How can you be in doubt? Here they are:

$$\Omega$$

Cantor looked at it and said:

CANTOR: But…  I see something else! Didn’t you forget about \(\Omega+1\)?

$$\mskip-200mu\{\emptyset,\,\{\emptyset\},\, \{\emptyset,
\{\emptyset\}\},\, \{\emptyset,\{\emptyset\}, \{\emptyset, \{\emptyset\}\}\},
\dots, \omega_0, \dots, \Omega\},$$

$$*\qquad*\qquad*$$

And it was all confusion.

Note:

In the creation myth God creates out of nothing. In today’s mathematics everything is reduced to sets that are constructed from nothing. Here we illustrate the Burali-Forti Paradox and von Newman’s definition of ordinal number. Kronecker said that the natural numbers were God’s creation and that everything else was man’s work. In the garden of Eden they were forbidden to eat from the tree of life in the middle of the garden and from the tree of the knowledge of good and evil. Cantor looking at \(\Omega\) transgresses this prohibition, thus paradise is destroyed, despite Hilbert’s words: No one will expel us from the paradise that Cantor has created for us.

In Genesis God rests on the seventh day. Here we do not see God resting, perhaps just for five minutes after he finishes the natural numbers.