<p>In this paper, we give characterizations of the set of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Pi ^1_{e+2}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Π</mi> <mrow> <mi>e</mi> <mo>+</mo> <mn>2</mn> </mrow> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation>-consequences, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Sigma ^1_{e+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Σ</mi> <mrow> <mi>e</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation>-consequences and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textsf{B}(\Pi ^1_{e+1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">B</mi> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="normal">Π</mi> <mrow> <mi>e</mi> <mo>+</mo> <mn>1</mn> </mrow> <mn>1</mn> </msubsup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-consequences of the axiomatic system of strong dependent choice for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Sigma ^1_i$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="normal">Σ</mi> <mi>i</mi> <mn>1</mn> </msubsup> </math></EquationSource> </InlineEquation> formulas <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Sigma ^1_i-\textsf{SDC}_0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Σ</mi> <mi>i</mi> <mn>1</mn> </msubsup> <mo>-</mo> <msub> <mi mathvariant="sans-serif">SDC</mi> <mn>0</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$e < i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>e</mi> <mo><</mo> <mi>i</mi> </mrow> </math></EquationSource> </InlineEquation>. Here, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textsf{B}(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the set generated by <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\wedge ,\vee ,\lnot $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>∧</mo> <mo>,</mo> <mo>∨</mo> <mo>,</mo> <mo>¬</mo> </mrow> </math></EquationSource> </InlineEquation> starting from <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>.</p>

On some subtheories of strong dependent choice

期刊论文
发表日期：

2026年7月11日

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Juan P. Aguilera,
Yudai Suzuki,
Keita Yokoyama

摘要

In this paper, we give characterizations of the set of $\Pi ^1_{e+2}$ $Π_{e + 2}^{1}$ -consequences, $\Sigma ^1_{e+1}$ $Σ_{e + 1}^{1}$ -consequences and $\textsf{B}(\Pi ^1_{e+1})$ $B (Π_{e + 1}^{1})$ -consequences of the axiomatic system of strong dependent choice for $\Sigma ^1_i$ $Σ_{i}^{1}$ formulas $\Sigma ^1_i-\textsf{SDC}_0$ $Σ_{i}^{1} - {SDC}_{0}$ for $e < i$ $e < i$ . Here, $\textsf{B}(\Gamma )$ $B (Γ)$ denotes the set generated by $\wedge ,\vee ,\lnot $ $\land, \lor, \neg$ starting from $\Gamma $ $Γ$ .