I have a phrase in English, and I need to determine if it is true or false. If it is true, I need to prove it, and if it is false, I need to disprove it. The phrase is based on the famous phrase "every pot has a lid", and it goes like this: "If there exist an infinite set of lids, then, all pots has a lid". As you can see, I have the -> connector here, along with the two quantifiers (all and exist). I am not sure how to prove or disprove it. Can you assist please ? Thank you in advance !
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OK, so if I understand you correctly, you are asking to prove or disprove the following argument or inference:
There exist an infinite number of lids
$\therefore$ All pots have a lid
Well, that one is easily shown to be invalid. I can have infinitely many lids, together with a pot without a lid. Indeed, why would pots have lids at all? Just because there are infinitely lids doesn't mean that there are lids everywhere. Consider:
There are infinitely many natural numbers
$\therefore$ All pots have a natural number written on it.
That does not follow, and your original argument is logically isomorph, so that one is not valid either.
Now, maybe you meant the following:
There exist an infinite number of lids
$\therefore$ For all pots there is a lid available (or: we can put a lid on every pot)
That still doesn't follow: even if there are infinitely many lids, that does not mean that every pot can be covered by a lid. If you haven't heard of the concept of 'cardinality', then I recommend you look up that concept. But the point is that there can be different 'kinds' of infinities, and that some infinities are 'greater' than others. Thus, for example, there are strictly 'more' real number than natural numbers, and they cannot be put into a one-to-one correspondence.
So, if you have as many lids as there are natural numbers, but you have as many pots as there are real numbers, then even though you have infinitely many lids, you still don't have enough to cover all of the pots!
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And what if the meaning was that every pot has a lid that fits it? Would it be valid then ? I tried writing predicated and created a truth table for ->, but it didn't lead me far. – user3275222 Apr 07 '17 at 18:41
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As for the natural number example, if I would like to write a number of every pot, it would be possible, since it's a one to one function, the number of pots is "equal" to the number of natural numbers – user3275222 Apr 07 '17 at 18:43
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@user3275222 I updated my Answer that I believe answer some of your follow-up questions. – Bram28 Apr 07 '17 at 19:00
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Thank you ! I understand what you are saying, it just that in my head both pots and lids are like natural numbers. I find it difficult to view the number of pots as the number of real numbers. Therefore to me it makes sense that it's "the same infinity". Yet, I still can't prove that every pot will have a lid. – user3275222 Apr 07 '17 at 19:06
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@user3275222: Even ignoring different cardinalities, just consider that you may have big pots and only small lids that don't fit, so infinitely many of these lids are useless. Even worse, you may have $2$ pots of the same size and only $1$ lid that fits and many other lids of smaller size, so infinitely many of them won't let you cover both pots! =) – user21820 Apr 17 '17 at 09:00