proof

consider the list of all prime numbers, ie. 2, 3, 5, 7, 11... now remove all but the first one digit prime, the first two digit prime, etc, leaving 2, 11, 101, etc. arrange these primes in a table, like the one below.

Code:

0002
0011
0101
1009
...

let us create a new number, N. let the first digit of N equal the first digit of the first number, plus one, the second digit equal the second digit of the second number, plus one, etc. with only the numbers on the table above, N = ...0223.

we have thus constructed a (probably) prime number that is different from every prime number on our list, proving that there are uncountably many primes, and, as a corollary, that there exists some prime number with infinitely many digits.

I probably don't know enough about primes, but,

What properties/laws are you using that makes N prime?

Have you also considered that, under the number system that allows infinite digit integers, since [...11111] is the largest number, there must also be a largest prime number?

If we constructed a table in that method that contains n primes, the resulting N has m digits. Surely there exists some n that produces a table that contains a number larger than N, if not containing N outright. The table that we used with n = infinity produces some prime N with some m digits; m in this case is presumably infinity. Does our infinite list contain a number larger than N? Which infinite value has more "infinite-ness", n or m?

consider the list of all possible sandwich toppings, i.e. Lettuce, tomato, basil, mustard... now remove all but the best topping, the second best topping, etc, leaving some damn fine toppings. Now arrange these toppings on an actual table, like the one below.

let us create a new sandwich, Steve. let the first topping on Steve be equal the first topping on the list, but better, The second topping be the second topping on the list, but better, etc. with only the toppings from the table above. Steve=Hella Good.

we have thus constructed a (probably) prime sandwich that is different from every prime topping on our list, proving that there are uncountably many sandwiches, and, as a corollary, that there exists some prime sandwich of infinite tastiness.

Um, Bob, you forgot the bread, now our table is infinitely messy.

True, but Hilbert over here says we still have room for another infinite set of toppings. So I'm not too worried about the mess.

issue: the list of all toppings is finite, but you included the 1st, 2nd, and so on best toppings. With no specified endpoint.

Thus Steve contains every topping, even the gnarly nasty ones. I will admit that you added each topping, but better, but how exactly can each of these toppings be any better than very bad?

• Glass
  
• Sand
  
• Cyanide
  
• Aerosol propellant
  
• Nintendo Switch cartridges
  
• Deodorant gel
  
• Pure, untreated Sulfur
  
• Neon green acryllic paint
  
• A wet lollipop covered in hair
  
• A really, really bad sandwich
  

Another problem: Steve can conceivably contain itself, and any sandwiches of infinite tastiness can conceivably contain themselves. Is there then a perfect sandwich that only contains toppings of itself, and, is it possible to create such a sandwich in the physical realm? If my hypothesis of "no" is correct, then why is life so cruel?

If sandwiches can contain themselves, then you wind up with the sandwich containing all sandwiches that don't contain themselves and only those sandwiches, and that's just not a road anyone wants to go down.

If all sandwiches are finite, then how can open sandwiches have no defined bottom layer?

If you put two pieces of bread on opposite sides of the world does that make the whole Earth a sandwich

What if every concieveable object was inside one human being's body, but they were the size of atoms? Would said human be a universe?

What does that have to do with sandwiches?

What do YOU have to do with sandwiches???

I ate one a little while ago.

Here's how - a human eats an everything sandwich, becomes a universe

But woulndn't the human just die if they ate the everything sandwich?

But woulndn't the human just die if they ate the everything sandwich?

it's an idealized abstraction.

yeah, it's what we call in the business,

a

"Question"