MHB Proving/Disproving "Every Pot Has a Lid

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The discussion revolves around the validity of the phrase "If there exist an infinite set of lids, then all pots have a lid." Participants express skepticism about the phrasing of the problem, arguing that it improperly mixes mathematical concepts with real-world relationships. Concerns are raised about the ambiguity and applicability of the problem, suggesting it should be reformulated using standard mathematical terminology. The conversation highlights the challenges of applying mathematical logic to scenarios that require knowledge of physical reality. Overall, the consensus is that the problem's construction is flawed and needs clarification for proper analysis.
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Hello all,

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 (every pot has a lid)".

As you can see, I have the "if-then" connector here (\implies), along with the two quantifiers (all and exist). I am not sure how to prove or disprove it.

On one hand, it sounds invalid, since having an infinite number of lids doesn't mean that there is no pot without a lid. On the other hand, the number of pots and number of lids are both natural numbers, and so the cardinality of both is equal, I think.

I was trying to write this down using predicates (using \forall \exists \implies \therefore), but couldn't do it.

Can you please assist in solving this problem ?

Thank you in advance !
 
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Yankel said:
"If there exist an infinite set of lids, then, all pots has a lid (every pot has a lid)".
If I were you, my first reaction would be to refuse to solve this problem. Making mathematical questions in such a way that they essentially involve knowledge about our world of physics, biology and human relationships is a poor way of constructing problems. Yes, problems for younger students often involve names and real situations, e.g., "A teacher announced that she is forming a student theater. Only senior students are eligible, and there has to be at least 14 members...". But such descriptions are readily translated into a purely mathematical model about sets, numbers, functions and so on. If one needs to know what the relationship between a pot and a lid is and how many pots are there in the world (countable and uncountable numbers are both impossible in our physical world), this is not good.

So I would ask you teacher to state the problem using regular mathematical vocabulary. If you have any guesses about the meaning of this phrase, feel free to say.
 
Thank you, I like your answer. I thought it was only me who thought this problem wasn't phrased properly.
 
Evgeny.Makarov said:
If I were you, my first reaction would be to refuse to solve this problem. Making mathematical questions in such a way that they essentially involve knowledge about our world of physics, biology and human relationships is a poor way of constructing problems. Yes, problems for younger students often involve names and real situations, e.g., "A teacher announced that she is forming a student theater. Only senior students are eligible, and there has to be at least 14 members...". But such descriptions are readily translated into a purely mathematical model about sets, numbers, functions and so on. If one needs to know what the relationship between a pot and a lid is and how many pots are there in the world (countable and uncountable numbers are both impossible in our physical world), this is not good.

So I would ask you teacher to state the problem using regular mathematical vocabulary. If you have any guesses about the meaning of this phrase, feel free to say.

Unfortunately some teachers/lecturers seem to love this sort of ambiguity. I would not be surprised if OP mentioned this to his prof. and got a response of "Oh yeah yeah of course, just assume bla bla and so and so"...
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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