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What are the nontrivial theorems one can prove about natural numbers which need transfinite induction? By "need transfinite induction" I mean one can show that the statement is not provable in arithmetic which only allows induction up to $\omega$. In particular, are there any statements which only state a claim about natural numbers but which require the use of infinite objects in their proof? The statement should not be metamathematical in nature (proof of consistency etc) or involve any ordinals or cardinals (i.e. infinite objects). Goodstein's theorem was mentioned as one such possibility in this MO questionthis MO question

What are the nontrivial theorems one can prove about natural numbers which need transfinite induction? By "need transfinite induction" I mean one can show that the statement is not provable in arithmetic which only allows induction up to $\omega$. In particular, are there any statements which only state a claim about natural numbers but which require the use of infinite objects in their proof? The statement should not be metamathematical in nature (proof of consistency etc) or involve any ordinals or cardinals (i.e. infinite objects). Goodstein's theorem was mentioned as one such possibility in this MO question

What are the nontrivial theorems one can prove about natural numbers which need transfinite induction? By "need transfinite induction" I mean one can show that the statement is not provable in arithmetic which only allows induction up to $\omega$. In particular, are there any statements which only state a claim about natural numbers but which require the use of infinite objects in their proof? The statement should not be metamathematical in nature (proof of consistency etc) or involve any ordinals or cardinals (i.e. infinite objects). Goodstein's theorem was mentioned as one such possibility in this MO question

edited to explain motivation
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Thinniyam Srinivasan Ramanatha
Thinniyam Srinivasan Ramanatha
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What are the nontrivial theorems one can prove about natural numbers which need transfinite induction? By "need transfinite induction" I mean one can show that the statement is not provable in arithmetic which only allows induction up to $\omega$. In particular, are there any statements which only state a claim about natural numbers but which require the use of infinite objects in their proof? The statement should not be metamathematical in nature (proof of consistency etc) or involve any ordinals or cardinals (i.e. infinite objects). Goodstein's theorem was mentioned as one such possibility in this MO question

What are the nontrivial theorems one can prove about natural numbers which need transfinite induction? The statement should not be metamathematical in nature (proof of consistency etc) or involve any ordinals or cardinals (i.e. infinite objects). Goodstein's theorem was mentioned as one such possibility in this MO question

What are the nontrivial theorems one can prove about natural numbers which need transfinite induction? By "need transfinite induction" I mean one can show that the statement is not provable in arithmetic which only allows induction up to $\omega$. In particular, are there any statements which only state a claim about natural numbers but which require the use of infinite objects in their proof? The statement should not be metamathematical in nature (proof of consistency etc) or involve any ordinals or cardinals (i.e. infinite objects). Goodstein's theorem was mentioned as one such possibility in this MO question

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Thinniyam Srinivasan Ramanatha
Thinniyam Srinivasan Ramanatha
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Transfinite induction vs induction in mathematics

What are the nontrivial theorems one can prove about natural numbers which need transfinite induction? The statement should not be metamathematical in nature (proof of consistency etc) or involve any ordinals or cardinals (i.e. infinite objects). Goodstein's theorem was mentioned as one such possibility in this MO question