4 Gödel, Hawking and the Foundations of Physics
63
“Yes!”.
“All right then, I’ll try to explain.”
Her mother pulled her armchair closer to Alice’s bed, rearranged the cushions, and settled to begin what she knew would
be a very long and complicated narrative. After just a few words,
however, when she next glanced down at her daughter, she
noticed that the little girl’s eyes were closed. Alice was fast
asleep.
4.2 The Price of Cakes in Gödel-Land
Our next goal is to try to explain what Alice’s mother could not, i.e. the
odd customs and rules of Gödel-land. However, first we need to abandon
the land of fairy stories, and enter into an altogether different abstract
dominion, the mathematical world of numbers. To begin, let us clarify what
we mean by a number. The reader may think this is unnecessary. Surely, a
number is just one of those quantities, like time and temperature, that everybody understands, but nobody seems quite able to define clearly (try asking
around).
Nevertheless, the concept is not really so difficult: if you have a collection
(or group) of any objects whatsoever and you disregard the nature of the
objects, the number is the only characteristic that remains of the group. For
instance, if you collect 2 pebbles, 3 cookies and one mango, and if you do
not care what the objects are (i.e., pebbles, cookies or mangoes), the only
thing that describes your collection is the number 6 (6 objects) …but don’t
eat the pebbles! The numbers, thus defined, are called integers (or natural
numbers), and are contained in one small portion of what we may call the
number world .
As trade began to flourish between humans, it became clear that the world
of integers was insufficient to encompass transactions when debt was incurred
by some individuals. In this case instead of owning 3 cookies in their collection, the debtors might owe 3 cookies to their neighbour’s collection, with the
two parties agreeing that the debt would be paid off some time in the future.
The number world was therefore extended by including negative numbers to
describe the debt. Much later, the number zero was included in the number
world. 1
1 It might be interesting to recall that the zero was “invented” about 600 CE by the Indian mathematician, Brahmagupta, but it was introduced to the Western world by Leonardo Fibonacci only in
1228 CE (in his Liber abbaci ).
63
“Yes!”.
“All right then, I’ll try to explain.”
Her mother pulled her armchair closer to Alice’s bed, rearranged the cushions, and settled to begin what she knew would
be a very long and complicated narrative. After just a few words,
however, when she next glanced down at her daughter, she
noticed that the little girl’s eyes were closed. Alice was fast
asleep.
4.2 The Price of Cakes in Gödel-Land
Our next goal is to try to explain what Alice’s mother could not, i.e. the
odd customs and rules of Gödel-land. However, first we need to abandon
the land of fairy stories, and enter into an altogether different abstract
dominion, the mathematical world of numbers. To begin, let us clarify what
we mean by a number. The reader may think this is unnecessary. Surely, a
number is just one of those quantities, like time and temperature, that everybody understands, but nobody seems quite able to define clearly (try asking
around).
Nevertheless, the concept is not really so difficult: if you have a collection
(or group) of any objects whatsoever and you disregard the nature of the
objects, the number is the only characteristic that remains of the group. For
instance, if you collect 2 pebbles, 3 cookies and one mango, and if you do
not care what the objects are (i.e., pebbles, cookies or mangoes), the only
thing that describes your collection is the number 6 (6 objects) …but don’t
eat the pebbles! The numbers, thus defined, are called integers (or natural
numbers), and are contained in one small portion of what we may call the
number world .
As trade began to flourish between humans, it became clear that the world
of integers was insufficient to encompass transactions when debt was incurred
by some individuals. In this case instead of owning 3 cookies in their collection, the debtors might owe 3 cookies to their neighbour’s collection, with the
two parties agreeing that the debt would be paid off some time in the future.
The number world was therefore extended by including negative numbers to
describe the debt. Much later, the number zero was included in the number
world. 1
1 It might be interesting to recall that the zero was “invented” about 600 CE by the Indian mathematician, Brahmagupta, but it was introduced to the Western world by Leonardo Fibonacci only in
1228 CE (in his Liber abbaci ).