4 ms·

> There are as many whole positive numbers as all fractions According to a specific non-colloquial definition of "as many".

adenadel 5y ago
Hm, I'm not so sure that just because the definition of a bijection is technical that it is not intuitive. I'll start an enumeration of the rationals 1 1/3 2 1/4 3 1/5 ... If you can prove that you can do this, is that really so non-colloquial? It is certainly what we mean by "as many" for all finite sets, so what is wrong with doing this for infinitely many sets?
- londgine 5y ago
  If every whole positive number is a fraction, but not every fraction is a whole positive number, then colloquially, I wouldn't define them as having "as many" elements as each other. Now, if you want to say they have the same cardinality (and you define cardinality as existing a bijection), then I would agree fully.
  - istjohn 5y ago
    What's worse, there are an infinite number of fractions that equal each whole number.
  - iovoid 5y ago
    Wouldn't there be exactly twice (or twice + 1, if you allow negative fractions) as much fractions, since fractions are represented as two positive numbers (plus a bit if you consider the sign). (The encoding could be "represent both numbers in binary, put the denominator in the odd bits (LSB = first bit), and the numerator in the even bits" so 2/3 => 10/11 => 1110 => 14)
  - foldr 5y ago
    The thing is that you could also use this kind of logic to show that there are more natural numbers than there are natural numbers. For example, you could associate 1 with all of the numbers from 1 to one million, and still have enough numnbers 'left' to associate each of 2,3,... with distinct natural numbers above one million.
unholiness 5y ago
My sister and I used to figure out who had more candy at halloween by lining up the pieces next to each other. The concept of bijection might be more intuitive than counting itself.
- cedilla 5y ago
  For all we know it's significantly older than counting. Pebbles representing bijections to wares like sheep (called calculi like in calculus) occur earlier than counting marks and much earlier than anything resembling numbers. There are still today human tribes that don't count at all.
  - tzs 5y ago
    > Pebbles representing bijections to wares like sheep (called calculi like in calculus) occur earlier than counting marks and much earlier than anything resembling numbers. How was this determined? I wouldn't expect that using pebbles this way would leave any distinctive marks or damage or residue on the pebbles that would allow an archaeologist several tens of thousands of years later to tell that was what the pebbles were used for.
kristjansson 5y ago
There’s an infinite number of each, so we’re already stretching colloquial definitions by comparing them

adenadel 5y ago

Hm, I'm not so sure that just because the definition of a bijection is technical that it is not intuitive. I'll start an enumeration of the rationals 1 1/3 2 1/4 3 1/5 ... If you can prove that you can do this, is that really so non-colloquial? It is certainly what we mean by "as many" for all finite sets, so what is wrong with doing this for infinitely many sets?

londgine 5y ago
If every whole positive number is a fraction, but not every fraction is a whole positive number, then colloquially, I wouldn't define them as having "as many" elements as each other. Now, if you want to say they have the same cardinality (and you define cardinality as existing a bijection), then I would agree fully.
- istjohn 5y ago
  What's worse, there are an infinite number of fractions that equal each whole number.
- iovoid 5y ago
  Wouldn't there be exactly twice (or twice + 1, if you allow negative fractions) as much fractions, since fractions are represented as two positive numbers (plus a bit if you consider the sign). (The encoding could be "represent both numbers in binary, put the denominator in the odd bits (LSB = first bit), and the numerator in the even bits" so 2/3 => 10/11 => 1110 => 14)
- foldr 5y ago
  The thing is that you could also use this kind of logic to show that there are more natural numbers than there are natural numbers. For example, you could associate 1 with all of the numbers from 1 to one million, and still have enough numnbers 'left' to associate each of 2,3,... with distinct natural numbers above one million.

unholiness 5y ago

My sister and I used to figure out who had more candy at halloween by lining up the pieces next to each other. The concept of bijection might be more intuitive than counting itself.

cedilla 5y ago
For all we know it's significantly older than counting. Pebbles representing bijections to wares like sheep (called calculi like in calculus) occur earlier than counting marks and much earlier than anything resembling numbers. There are still today human tribes that don't count at all.
- tzs 5y ago
  > Pebbles representing bijections to wares like sheep (called calculi like in calculus) occur earlier than counting marks and much earlier than anything resembling numbers. How was this determined? I wouldn't expect that using pebbles this way would leave any distinctive marks or damage or residue on the pebbles that would allow an archaeologist several tens of thousands of years later to tell that was what the pebbles were used for.

kristjansson 5y ago

There’s an infinite number of each, so we’re already stretching colloquial definitions by comparing them