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We give two proofs of ${\\sf{ERT}}$ in ${\\sf{RCA}}_0$, using different methods to eliminate the use of ${\\Sigma^0_2}$ induction. Working in the weakened base system ${\\sf{RCA}}_0^*$, we prove that ${\\sf{ERT}}$ is equivalent to ${\\Sigma^0_1}$ induction and ${\\sf{ECT}}$ is equivalent to ${\\Sigma^0_2}$ induction. We conclude with a Weihrauch analysis of the principles, showing ${\\sf{ERT}} {\\equiv_{\\rm W}} {\\sf{LPO}}^* {<_{\\rm W}}{{\\sf{TC}}_{\\mathbb N}}^* "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1812.09943","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2018-12-24T15:51:00Z","cross_cats_sorted":[],"title_canon_sha256":"70b69bd26809225a46b9812a59f2141eedbaa92640987ba51be23ec6b2f87e00","abstract_canon_sha256":"2b34a7cee3c203ce351e1b169cb1ed19fa09d50f9355a122cfe0ef3210f863fc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:57:26.302010Z","signature_b64":"NBT65vUjjP/yvgXeIIGD91NSmSLxFwq+VDI5N2YODjmMswNVlP7Eh6AcOJh+x070G6QjM2g1HE5MSlMEZEhCDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"99a68f834143d6c6b681c7e83f28d13f01ff76992f30e868137cecab4677f25a","last_reissued_at":"2026-05-17T23:57:26.301435Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:57:26.301435Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Combinatorial principles equivalent to weak induction","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Arno Pauly, Caleb Davis, Denis R. Hirschfeldt, Jake Pardo, Jeffry L. Hirst, Keita Yokoyama","submitted_at":"2018-12-24T15:51:00Z","abstract_excerpt":"We consider two combinatorial principles, ${\\sf{ERT}}$ and ${\\sf{ECT}}$. Both are easily proved in ${\\sf{RCA}}_0$ plus ${\\Sigma^0_2}$ induction. We give two proofs of ${\\sf{ERT}}$ in ${\\sf{RCA}}_0$, using different methods to eliminate the use of ${\\Sigma^0_2}$ induction. Working in the weakened base system ${\\sf{RCA}}_0^*$, we prove that ${\\sf{ERT}}$ is equivalent to ${\\Sigma^0_1}$ induction and ${\\sf{ECT}}$ is equivalent to ${\\Sigma^0_2}$ induction. 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