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Think of a city, say the capital of Germany, Berlin. Now I can easily come up with another city, like San Fransisco. Now with two cities in mind, I can still come up with another city (Stockholm, for example). By induction there should be infinite many cities because I can clearly come up with yet another city no matter how many cities I have thought about.

This is obviously wrong. What exactly am I doing wrong here?

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Cities, unlike natural numbers, are not defined by induction. The notion of a "next city" or the "successor city" isn't well-defined. Without that function, which would capture the inherent order of the cities, you can just have two cities A and B. If you have A in mind, you can find another city B. If you have B in mind, you can find another city A. It is true that whatever city you have in mind you can find another city. You get the idea. –  Hunan Rostomyan Apr 19 at 17:27
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"Think of a day of the week, say Monday. Now I can easily come up with another day of the week, Tuesday, and still another, Wednesday". OK. "I can clearly come up with yet another day of the week no matter how many days I have thought about". Wait, what? –  Steve Jessop Apr 19 at 18:09
    
You are not doing anything wrong you are just not doing MATHEMATICAL induction. –  Asphir Dom Apr 19 at 23:13
    
Can you really think of 1000 cities, all at the same time? I know I can't, even though I know there are at least that many. –  David Wallace Apr 20 at 10:02
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5 Answers 5

For induction you need to define a rule of how to go from n to n+1. You fail to define a rule of how to go from "Germany, Berlin" to "Germany, Berlin, San Fransisco".

The rule could be: Add "San Fransisco" to the set. Of course you can't use that rule to prove that there's an infinitive amount of cities.

Your rule could be: Add a city that's not already in the list. That rule presupposes that there always a city that's not on the list. Basically you would assume what you want to prove.

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"Basically you would assume what you want to prove". Indeed you assume exactly what you want to prove. Infinity can be characterized as, "whatever number I have, I can get another one". That's because "finite" can be characterized (with care) as "there's a number that, if I have that many, there are no more", and infinity is the opposite. –  Steve Jessop Apr 19 at 18:12
    
Christian, yes, but this only applies to mathematical induction, not inductive arguments more generally. –  ChristopherE Apr 19 at 19:44
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Your inductive step is

Given a list of cities, I can always give a new city that is not on the list.

and I would say this is false. It is just not true that one

can clearly come up with yet another city no matter how many cities I have thought about.

As this premise is false the induction does not give a the correct answer (though the argument itself is valid).

Your question is related to the widely known problem of induction. But this case is less problematic because one can actually list all cities and see that the list is indeed finite. If the task was to show that it is infinite it would be a different matter.

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For example, if your definition of "city" includes that it be populated, then there is no way you will come up with more than 8 billion cities. –  Nate Eldredge Apr 20 at 0:52
    
@NateEldredge: At least, assuming that the definition also includes cities being disjoint. –  Ilmari Karonen Apr 20 at 4:26
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Your induction draws from a set of city names. However, you have failed to show a proof that there is an infinite number of city names (and that all those cities exist).

This is like saying...

  1. I ate one potato chip from the bag.
  2. I ate another potato chip from the bag.
  3. There are still more potato chips in the bag.
  4. Therefore there are INFINITE POTATO CHIPS!

Another way to look at it, your argument is circular. Your inductive proof that there is an infinite set of cities assumes there is an infinite set of cities to draw from.

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The "axiom of induction" regarding numbers says that :

for every property P of numbers, if P holds for 0 and for every number n, if P holds for n, then P holds also for n+1 (the "number after n"), then P will hold for every number.

This axiom of induction exploit the "structure" of the set of natural numbers : there is a first one, and they have an "order" which, for every numbers gives us the "next one".

Per se, induction does not implies the infinity of numbers.

The infinity of them is licensed by other axiom; in detail, by the axiom which states that there is a number (usually 0) which is not the "next one" of any number and by the axiom which states that there are no two different numbers which has the same "next one".

In your case, the "collection" of all cities does not have the structure of natural numbers (a first one, and an "order" that for every numbers gives us the "next one"); thus, we have no reason to assume that there exist an infinite number of cities.

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Mauro, Yes, but as I commented to Christian also, this applies only to mathematical induction, not to inductive arguments more generally. The induction in the question fails not only as mathematical induction but also for more general reasons. –  ChristopherE Apr 19 at 19:47
    
@ChristopherE - but I do not think the above argument (apart being wrong) has the "form" of an inductive argument as per indcutive logic (Carnap, Hintikka, etc.). Inductive logic is aimed at formalizing arguments establishing generalities (law of nature) from observed particular case. The above argument is aimed at establishing the "existence" of an inifinite number of objects. No inductive argument whatever can support an existence claim of this sort. –  Mauro ALLEGRANZA Apr 20 at 8:35
    
Yes, I think you're right about that. –  ChristopherE Apr 20 at 13:19
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Your assumption that given a list of cities you can always find one more city, is false. In fact at some point you will have a list of cities, but you will not be able to name one more city.

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