# Many Numbers

On a dark, winter evening, a mathematician was walking down back home, with a baguette that she just bought from the grocery store. Nearing the house, she saw a small figure huddled under the streetlamp, shivering in the cold. On approaching close, the mathematician recognized the figure as being 0, so she called out, “Hey 0! What are you doing out here, all alone in the cold by yourself? You’ll get frostbite if you stay here all night.”

0 replied, “Well, I don’t know anybody, so I don’t have anywhere to go.”

The mathematician was puzzled. She was already feeling cold from standing still. Being a kind spirit, she made an offer, “Why don’t you come over to my place for the night?”

0 happily agreed to the suggestion. So they headed back to the mathematician’s house.

Removing her jacket, the mathematician went into the kitchen and lit the stove to make some hot chocolate.

0 walked in slowly behind, and took up one of the small chairs. The mathematician did not like small talk, so she did not make any. 0 just stared dreamily through the kitchen window. In a short while, the hot chocolate was ready. The mathematician poured it into a couple of mugs and set them on the quaint dining table, “There we go.” Just as she was about to take a seat herself, the doorbell rang. “Well, that’s strange … I wonder who it could be at this time of the night,” she thought.

As she opened the door, she recognized 1. 1 spoke very hurriedly, “0 was the only number I know and 0 was not outside … I must be with 0.” The mathematician was a bit perplexed but she let 1 come in.

As 1 sat on a chair beside 0, the mathematician saw that one of the hot chocolate mugs had been emptied, rinsed and was drying on the kitchen rack. She was slightly taken aback, but the poor sight of 1 still shivering from the cold made her feel bad, so she said, “Hey 1. I have a mug of hot chocolate left, you could have a half of it if you like.” 1 gratefully accepted the offer, so she split the one mug of hot chocolate equally into the two mugs.

Just as she was about to take a seat, the doorbell rang again. As she opened the door, she was not surprised as 2 spoke in a hurried fashion, not unlike 1, “Hey, could you take me in … ? 0 and 1 are the only two numbers I know and they weren’t outside … I must be with them.” The mathematician rolled her eyes, “Yeah, come in.” She went in, split the hot chocolate from the half-full mug into two, handed one of the mugs to 2, and headed back to the door. As she opened the door, she saw 3 taken aback slightly. She said, “Oh, come on in, I was expecting you.” Again she went in, split the hot chocolate into two and headed back to the door. She was getting a bad feeling that there wouldn’t be any left for her in a while.

True enough, after a certain while, her house was crowded with numbers. As she handed out the mug to 76 – who had just come in – 76 exclaimed, “But there isn’t even an atom of hot chocolate in here!” The mathematician was pissed, “Well, you can take it or leave it. Having some partial atom is probably better than the empty mug that will be left in the end for me.” 76 was astonished by her rudeness; nonetheless, the mug handed by her was promptly emptied, rinsed and put back to dry.

Despite there being an ever decreasing amount of hot chocolate, the numbers didn’t stop coming. They always had the same excuse, “I only know so and so and none of them are outside … I must be with them.” After a certain while, there were just too many numbers to fit in the house, so the mathematician started putting them all inside her trusty little laptop, so that she wouldn’t lose them. However, even the laptop ran out of storage space after a while. After thinking for a while, the mathematician came up with a clever solution; she pulled out her notebook and wrote the following:

```
Number | Knows
0 | Nobody
1 | 0
2 | 0,1
... | ...
```

She did not want to write down all of them
– otherwise the paper would run out of space with fewer numbers than the computer –
so she used the `...`

to represent that she knew and understood the rest.
At the moment she finished writing this down,
the remaining little hot chocolate vanished and the doorbell stopped ringing.

The mathematician was tired. She hadn’t had anything to eat or drink since getting back and now the hot chocolate was gone. She did not want to go to bed on an empty stomach; so she sliced the baguette in half, took out a jar of marmalade from the shelf above the kitchen counter, and took a seat at the now quiet dining table. As she was about to start spreading the marmalade, the doorbell rang again.

Surely she must be hallucinating. Who else could it be at this time of the night? The doorbell rang again. She wearily got up. As she curiously opened the door, a strange figure looked at her with questioning eyes, “0, 1, 2 etc. are the only numbers I know and they are not outside … I must be with them.” The mathematician was puzzled.

“Do you have a name?”

“ω.”

“Okay ω. Come on in.”

So ω was given a seat and was offered half a baguette. At that moment, the doorbell rang again. Another conversation at the door followed.

“I only know 0,1,2 etc. and ω and they weren’t outside … I must be with them.”

“Do you have a name?”

“ω + 1.”

The figure was invited in and offered a quarter of the baguette. After that, the mathematician went back to her notebook and wrote the following:

```
Number | Knows
0 | Nobody
1 | 0
2 | 0,1
... | ...
ω | 0,1,...
ω+1 | 0,1,... ω
ω+2 | 0,1,... ω,ω+1
... | ...
```

Just as she finished writing, the remaining quarter of the baguette vanished in a poof! And yet, soon enough, the doorbell rang again. Having been overly generous with all her visitors, the mathematician did not have any food left in the pantry. Turning away the next visitor seemed too embarrassing, so she lingered around the dining table, thinking. After a few seconds she drew a baguette:

```
___________
(_/_/_/_/_/_)
```

Then she added a line to her table:

```
Number | Knows
0 | Nobody
1 | 0
2 | 0,1
... | ...
ω | 0,1,...
ω+1 | 0,1,... ω
ω+2 | 0,1,... ω,ω+1
... | ...
ω+ω | 0,1,... ω,ω+1,...
```

As if by magic, the baguette that she had drawn suddenly shrunk by half and it now looked like this:

```
______
(_/_/_/
```

Soon enough, the doorbell rang again. This time, she added two more lines to her table.

```
Number | Knows
0 | Nobody
1 | 0
2 | 0,1
... | ...
ω | 0,1,...
ω+1 | 0,1,... ω
ω+2 | 0,1,... ω,ω+1
... | ...
ω+ω | 0,1,... ω,ω+1,...
ω+ω+1 | 0,1,... ω,ω+1,... ω+ω
ω+ω+2 | 0,1,... ω,ω+1,... ω+ω,ω+ω+1
... | ...
```

And poof! The remaining half baguette vanished too. The mathematician knew what to do now. Every time that the doorbell would ring, it would be sufficient to draw one more baguette and add four more lines to the table. But how long could she keep on drawing baguettes? She had to get some sleep and get back to work in the morning. So she thought, “Why don’t I draw all the baguettes at once?”

```
___________ ___________ ___________
(_/_/_/_/_/_) (_/_/_/_/_/_) (_/_/_/_/_/_) ...
```

She also added three more lines to her table.

```
Number | Knows
0 | Nobody
1 | 0
2 | 0,1
... | ...
ω | 0,1,...
ω+1 | 0,1,... ω
ω+2 | 0,1,... ω,ω+1
... | ...
ω+ω | 0,1,... ω,ω+1,...
ω+ω+1 | 0,1,... ω,ω+1,... ω+ω
ω+ω+2 | 0,1,... ω,ω+1,... ω+ω,ω+ω+1
... | ...
. | .
. | .
. | .
```

“Surely, that’s the end of it!”
Promptly enough, all the drawn baguettes vanished with a poof!
Even the `...`

at the end vanished.

By now, she was too exhausted to even change her clothes; she just hopped into bed. Snuggling between the blankets, she adjusted the pillow and set the alarm for the next day, unsure whether she would be able to wake up on time. As she was able to close her eyes, the doorbell rang again.

“Oh no! Not again!!!”

She limbered out of the bed and opened the door, her eyes still half-closed. The strange figure at the door spoke without being asked.

“I’m Ω.”

“Come in Ω. Would you like a baguette with some marmalade?”

“Thank you.”

“Oh, I haven’t even begun yet.”