Then let’s take the approach of asking your intuition to answer different questions. Start with the one about A and P in the edit to my comment above.
The idea is to make you feel the contradiction in your intuitive decisions, which helps discard the intuition as not useful in the domain of infinite sets. Then you’ll have an easier time learning about the various mathematical approaches to the problem because you’ll feel that there is a problem.
possibly one thing that made the intuitive-ness of the primes vs. naturals problem is that the naturals is a special set (both mathematically and intuitively). How would you compare P={primes} and A=P+{even numbers}? A still strictly contains P, so intuitively it’s “bigger”, but now you can’t say that every number in A “accounts” for itself if you want to build a mapping from A to the naturals (i.e. if you want to arrange A as a series with indexes).
A seems to contain the number 2 twice. Is that on purpose?
Question 2: how do you compare the even numbers and the odd numbers? Intuitively there is the same amount of each. But neither is a subset of the other.
In that case the sensible thing to do seems to me to pair every number with one adjacent to it. For instance, one can go with two, and three can go with four, etc.
A seems to contain the number 2 twice. Is that on purpose?
In the mathematical meaning of a ‘set’, it can’t contain a member twice, it either contains it or not. So there’s no special meaning to specifying 2 twice.
In that case the sensible thing to do seems to me to pair every number with one adjacent to it. For instance, one can go with two, and three can go with four, etc.
Here you’re mapping the sets to one another directly instead of mapping each of them to the natural numbers. So when does your intuition tell you not to do this? For instance, how would you compare all multiples of 2 with all multiples of 3?
Half of the multiples of 3 are also multiples of 2. Those can map to themselves. The multiples of 3 that are not also multiples of 2 can map to even numbers between the adjacent two multiples of 3. For instance, 6 maps to itself and 12 maps to itself. 9 can map to a number between 6 and 12; let’s pick 8. That leaves 10 unaccounted for with no multiples of 3 going back to deal with it later; therefore, there are more multiples of 2 than of 3.
Essentially, you’ve walked up the natural numbers in order and noted that you encounter more multiples of 2 than multiples of 3. But there’s no reason to privilege that particular way of encountering elements of the two sets.
For instance, instead of mapping multiples of 3 to a close multiple of 2, we could map each multiple of 3 to two-thirds of itself. Then every multiple of 2 is accounted for, and there are exactly as many multiples of 2 as of 3. Or we could map even multiples of 3 to one third their value, and then the the odd multiples of 3 are unaccounted for, and we have more multiples of 3 than of 2.
Your intuition seems to correspond to the following: if, in any large enough but finite segment of the number line, there are more members of set A than of set B; then |A|>|B|.
The main problem with this is that it contradicts another intuition (which I hope you share, please say so explicitly if you don’t): if you take a set A, and map it one-to-one to a different set B (a complete and reversible mapping) - then the two sets are equal in size. After all, in some sense we’re just renaming the set members. Anything we can say about set B, we can also say about set A by replacing references to members of B with members of A using our mapping.
But I can build such a mapping between multiples of 2 and of 3: for every integer x, map 2x to 3x. This implies the two sets are equal contradicting your intuition.
Then let’s take the approach of asking your intuition to answer different questions. Start with the one about A and P in the edit to my comment above.
The idea is to make you feel the contradiction in your intuitive decisions, which helps discard the intuition as not useful in the domain of infinite sets. Then you’ll have an easier time learning about the various mathematical approaches to the problem because you’ll feel that there is a problem.
A seems to contain the number 2 twice. Is that on purpose?
In that case the sensible thing to do seems to me to pair every number with one adjacent to it. For instance, one can go with two, and three can go with four, etc.
In the mathematical meaning of a ‘set’, it can’t contain a member twice, it either contains it or not. So there’s no special meaning to specifying 2 twice.
Here you’re mapping the sets to one another directly instead of mapping each of them to the natural numbers. So when does your intuition tell you not to do this? For instance, how would you compare all multiples of 2 with all multiples of 3?
Half of the multiples of 3 are also multiples of 2. Those can map to themselves. The multiples of 3 that are not also multiples of 2 can map to even numbers between the adjacent two multiples of 3. For instance, 6 maps to itself and 12 maps to itself. 9 can map to a number between 6 and 12; let’s pick 8. That leaves 10 unaccounted for with no multiples of 3 going back to deal with it later; therefore, there are more multiples of 2 than of 3.
Essentially, you’ve walked up the natural numbers in order and noted that you encounter more multiples of 2 than multiples of 3. But there’s no reason to privilege that particular way of encountering elements of the two sets.
For instance, instead of mapping multiples of 3 to a close multiple of 2, we could map each multiple of 3 to two-thirds of itself. Then every multiple of 2 is accounted for, and there are exactly as many multiples of 2 as of 3. Or we could map even multiples of 3 to one third their value, and then the the odd multiples of 3 are unaccounted for, and we have more multiples of 3 than of 2.
Your intuition seems to correspond to the following: if, in any large enough but finite segment of the number line, there are more members of set A than of set B; then |A|>|B|.
The main problem with this is that it contradicts another intuition (which I hope you share, please say so explicitly if you don’t): if you take a set A, and map it one-to-one to a different set B (a complete and reversible mapping) - then the two sets are equal in size. After all, in some sense we’re just renaming the set members. Anything we can say about set B, we can also say about set A by replacing references to members of B with members of A using our mapping.
But I can build such a mapping between multiples of 2 and of 3: for every integer x, map 2x to 3x. This implies the two sets are equal contradicting your intuition.