December 15, 2022

Dear Nikola,

Any time you want to visit me, let please give me an opportunity to confirm by telephone or by e-mail that the kitchen door is open. You are ALWAYS welcome.

Thank you for pointing me to Stephen Cole Kleene. Introduction to Metamathematics.  I have spent some time reading in this book.  I have spent more time thinking about its messages, among them the existence of one or more, perhaps of an infinity of infinite sets.  I have my own thoughts about these infinities, thoughts on which I doubt that Stephen Cole Keene would give me a passing grade.

I turn to Georg Cantor for answers.  I find Cantor's "Mengenlehre" translated into English as "Set Theory", thus "setting up" the contradiction out of which academicians have made a thriving business. In colloquial German "eine Menge" is an uncounted accumulation, which being uncounted is therefore numerically unlimited or infinite.  

As my witness I call Martin Luther who translated Genesis 16:10, καὶ εἶπεν αὐτῇ ὁ ἄγγελος κυρίου πληθύνων πληθυνῶ τὸ σπέρμα σου καὶ οὐκ ἀριθμηθήσεται ἀπὸ τοῦ πλήθους ich wil deinen samen also mehren, das er fur groszer menge nicht sol gezelet werden. 1 Mos. 16, 10 10 And the angel of the Lord said unto her, I will multiply thy seed exceedingly, that it shall not be numbered for multitude.  I note that the Greek "οὐκ ἀριθμηθήσεται ἀπὸ τοῦ πλήθους" might colloquially be translated "shall not be arithmetized on account of its multiplicity."  

In colloquial English, if I understand correctly, "a set" is a numerically limited collection of items.  The multiplicity implicit in infinity is an entangled quantum phenomenon which entails as its uncertainty principle "uncountability", a state which precludes counting, just as the Heisenberg Uncertainty precludes simultaneous measurement of place and momentum.  The infinity entanglement collapses into a finite series of cardinal numbers as soon as one begins counting, by adding to infinity or by subtracting from it.  The terms "infinity of numbers" or "numerical infinity" are "contradictiones in adjecto" since numbers, being "quanta" are in essence, bounded, are inherently finite. 

Søren Kierkegaard famously declared, subjectivity is the truth.  If he was correct then infinity might be defined by what I cannot imagine myself doing, and that in turn defined by my statistical life expectancy of 3.5 years. Assuming from now on insomnia, and unfettered stamina, at one digit per second, infinity would "begin" after I had completed counting to 110376000. Or if calibrated by cycles per second of my continuous singing until I die, of the note A (440 Hertz), 48565440000. Or if I counted the paces of one foot each, which I take at the rate of one step per second, .6818 miles per hour, 110376000 steps for a total of 20903 miles.  These numbers suggest the boundary of numbers meaningful for me.  That's where "my" infinity might "begin". 

Hilberts Hotel https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel has an infinite number of individual guests housed in an infinite number of rooms. However, it is always possible to accommodate yet another guest by moving the occupant of room No. 1 to room Number Infinity plus One, assumping one is willing to countenance precipitating the collapse of the previously existing infinity into a finite series of cardinal numbers.
EJM