A new model construction by making a detour via intuitionistic theories II: Interpretability lower bound of Feferman's explicit mathematics T0

We partially solve a long-standing problem in the proof theory of explicit mathematics or the proof theory in general. Namely, we give a lower bound of Feferman’s system T0 of explicit mathematics (but only when formulated on classical logic) with a concrete interpretat ion of the subsystem Σ12-AC+ (BI) of second order arithmetic inside T0. Whereas a lower bound proof in the sense of proof-theoretic reducibility or of ordinalanalysis was already given in 80s, the lower bound in the sense of interpretability we give here is new. We apply the new interpretation method developed by the author and Zumbrunnen (2015), which can be seen as the third kind of model construction method for classical theories, after Cohen’s forcing and Krivine’s classical realizability. It gives us an interpretation between classical theories, by composing interpretations between intuitionistic theories

Decision response times (RT) when choosing RO (left) and SO (right) in <i>Toxoplasma</i>-infected and <i>Toxoplasma</i>-free subjects.

<p>The graph shows arithmetic means, standard errors and a p-value.</p

Risk task: A sample screenshot from the study.

<p>The two numbers on the left represent the gamble’s possible gain and loss amounts (<i>Top</i> and <i>Bottom</i>, respectively). The number on the right represents the guaranteed amount. Participants had to indicate which option they wanted to choose by clicking the corresponding button.</p

Regression analysis of individual parameters.

<p>Coefficients in all columns are logistic regression estimates, clustered standard errors are in parentheses;</p><p>*** indicate significance at 1% level.</p><p>* indicate significance at 10% level.</p><p>^a<i>Toxoplasma</i> is a dummy variable and equals 1 for <i>Toxoplasma</i>-infected subjects.</p><p>^bRhD is a dummy variable and equals 1 for RhD positive subjects.</p><p>Regression analysis of individual parameters.</p

Summary statistics for the sample.

<p>^a P shows statistical significance for two tailed t test.</p><p>Summary statistics for the sample.</p

Frequency of risky choices when the expected value of RO is less than SO (left) and the expected value of RO is greater than SO (right) in <i>Toxoplasma</i>-infected and <i>Toxoplasma</i>-free subjects.

<p>The graph shows arithmetic means, standard errors and a p-value.</p

Logistic regression.

<p>Coefficients in all columns are logistic regression estimates, clustered standard errors are in parentheses;</p><p>*** indicate significance at 1% level.</p><p>^a<i>Toxoplasma</i> is a dummy variable and equals 1 for <i>Toxoplasma</i>-infected subjects.</p><p>^bRhD is a dummy variable and equals 1 for RhD positive subjects.</p><p>Logistic regression.</p

Mixed-effects linear regression.

<p>Coefficients in all columns are logistic regression estimates, clustered standard errors are in parentheses;</p><p>*** indicate significance at 1% level.</p><p>^a<i>Toxoplasma</i> is a dummy variable and equals 1 for <i>Toxoplasma</i>-infected subjects.</p><p>^bChoice is a dummy variable and equals 1 if subjects chose risky option.</p><p>^cRhD is a dummy variable and equals 1 for RhD positive subjects.</p><p>Mixed-effects linear regression.</p

Summary statistics of the parametric data.

<p>^aP shows statistical significance for two tailed t-test.</p><p>Summary statistics of the parametric data.</p

Frequency of risky choices in <i>Toxoplasma</i>-infected and <i>Toxoplasma</i>-free subjects.

<p>The graph shows arithmetic means, standard errors and a p-value.</p