Inductive Reasoning: Si to Sn & i→∞

[Gear300]

Considering that the truth of S_i implies the truth of S_i+1, i ε natural numbers, then starting from the truth of S₁, one can state the truth of S_n, n being any natural number. I was wondering whether we can make any statement as i→∞, such that the limS_i as i → ∞ is also true?

 

[Stephen Tashi]

as i→∞, such that the limS_i as i → ∞ is also true?

I'm tempted to say "No" immediately because one interpretation of what you mean is "if a statement is true for any given natural number then it is true for the entire set of natural numbers". That wouldn't be valid reasoning. For example: S_i = "There is a natural number greater than i" versus "There is a natural number greater than any number in the set of natural numbers".

What you said mentions: " $lim_{n \rightarrow \infty} S_i"$ and you didn't define what this means. If we take "1" as representing "true" and interpret the limit as a the limit of a numerical sequence, then the limit is 1. But that is a different interpretation than in the previous paragraph.

 

[Gear300]

If we were to check the countability of the cartesian product between two countable sets, A = {a1, a2, a3, ...} and P = {p1, p2, p3, ...}, the resultant set should be countable: If were to consider
{(a1,p1), (a1,p2), (a1,p3), ...} = M₁
{(a2,p1), (a2,p2), (a2,p3), ...} = M₂
{(a3,p1), (a3,p2), (a3,p3), ...} = M₃
... ... ... ... ... ... ... ... ... ... ... ... ...
Taking the union of all the M_i's results in taking the union of a countable number of countable sets, and so the cartesian product A x P is countable. From this reasoning, we could say that for N = set of natural numbers, Nⁿ for some finite n is countable:
Nⁱ x N = Nⁱ⁺¹
N¹ x N = N² ---> since N is countable, so is N²
N² x N = N³ ---> since N² is countable, so is N³, and so forth.
However, if we were to continue this for Nⁿ as n→∞ (an infinite dimensional space), the countability is arguable (I initially thought that it would be countable by inductive reasoning):
Consider the set of real numbers [0,1). This set has power c of the real numbers. We can write each number as some decimal expansion 0.a1a2a3...
We can map these numbers into the infinite dimensional case of Nⁿ by making the correspondence 0.a1a2a3... → (a1,a2,a3,...). [0,1) would then be mapped to a proper subset of Nⁿ (since each decimal place is bounded between 0 and 9, and forms ending in 99999... converge to some other form). Therefore Nⁿ can not be countable for the infinite dimensional case.
I had thought that induction implied that you could apply the condition on S_i onto limS_i as i → ∞, so I am unsure of whether there is a flaw in this proof somewhere.

 

[Stephen Tashi]

However, if we were to continue this for Nn as n→∞

That phrase has an intuitive appeal, but to reason precisely about it you have to give it a precise definition. What does it mean? After all, we can say that the limit of a numerical sequence is the result of "continuing a process to infinity", but you can't reliably use that definition in proofs. You have to use the epsilon-delta definition.

 

[CellCoree]

Inductive reasoning is a powerful tool in the scientific method, as it allows us to make generalizations and predictions based on observed patterns. In this case, the content suggests that if Si is true, then Si+1 will also be true, and this pattern continues for all natural numbers. This is a valid use of inductive reasoning, as it is based on a logical progression and can be tested and verified.

However, when considering the limit as i approaches infinity, we must be cautious. While it may seem intuitive to assume that the pattern will continue indefinitely, this is not always the case. There could be factors or variables that we are not considering, which could change the outcome as i approaches infinity.

Therefore, while we can make a statement about Si being true for all natural numbers, we cannot make a definitive statement about the limit of Si as i approaches infinity being true. It is important to continue testing and gathering evidence to support this conclusion, rather than assuming it to be true based solely on inductive reasoning.